Notation, in simple word means any symbol which denotes a particular process of calculation. For instance, if we want to add any 2 numbers like 5 and 6, then we will have to use the notation for addition, which is +. Thus, notation for addition is " + ".
The above picture shows the basic 4 mathematical notations on which the whole concept of mathematics depends.

The formal definition of notation is defined as follows:

Notation is a symbol used for representing numbers, equations, elements or any mathematical sign or character, which has a particular meaning that conveys the procedure for solving any problem.

For example, if we want to solve a problem of adding 3 and 4, we can simply put the notation of addition, like 3 + 4 = 7, where '+' conveys the meaning that addition has to be performed. So, + symbol is used for the procedure of addition. Hence, in short, Notation is a symbol used for the procedure to solve any problem.

The following is a list of some common and basic notations used in mathematics:

Notation  Meaning of the symbol  Example 
 f(x) Symbol for s function of x  f(x) = 6x + 2 
 (f o g) Symbol for the composition of the function
denoted as (f o g)(x) = f(g(x)) 
f(x) = 6x, g(x) = x - 2
then, (f o g)(x) = 6 (x - 2)
(a, b)  Symbol for open interval denoted as $(a,b) = {x:a$x\epsilon (4, 10)$
[a, b] Symbol for closed interval denoted as $[a,b] = {x:a\leq x\leq b}$
$x\epsilon [4,10]$
x  Symbol for the variable x, which is always an unknown quantity If 6x = 18, then x = 3 
$\equiv $  Symbol for equivalence  Like, 3 is identical to 3
$\sim $ Symbol for approximately equal to, which is also known as weak approximation 15$\sim $14.432
$\approx $  Symbol for approximately equal to, which is proper approximation
sin(0.01) $\approx $ 0.01 
$\left | A \right |$  Symbol for determinant of matrix A determinant of matrix A
det(A) Another symbol for determinant of matrix A determinant of matrix A
$\left \| x \right \|$  Symbol for the Norm of a variable  $\left \| x \right \|$ = x + 2 
BT Symbol for transpose of a matrix B $(B^{T})_{ij}=(B)_{ji}$
$B^{\div }$  Symbol for the conjugate transpose of a matrix  B
$(B^{\div })_{ij}=(B)_{ji}$ 
B* Another symbol for the conjugate transpose
of a matrix B
B-1 Symbol for the inverse of a matrix B BB-1 = I
rank (B) Symbol for the rank of a matrix B  rank (B) = 3
dim (B) Symbol for the dimension of a matrix B  rank (B) = 3

In simple words, in mathematics, a notation is used as a short form by writing a symbol or sign, for conveying the meaning of a long sentence in short. For example, if we want to write, that all the numbers from 0 to 20 are included, then instead of writing this whole sentence, we can just write it using the notation of closed parenthesis [ ] as follows:

Notation Definition
Hence, this notation of [0, 20] automatically conveys the meaning that all numbers from 0 to 20 are included, and on the other hand, if we want to write the numbers only above 0 and below 20, then we will simply use the notation of open parenthesis () as shown above, i.e., (0, 20).

Notations can be classified in accordance with, in which field of mathematics they are applied. Therefore, various types of notations can be as basic mathematical notations, geometric notations, algebraic notations, probability and statistical notations, set theory notations, logical notations, calculus and analysis notations, notations for the representation of system of numbers, notations of Greek symbols used for representing angles, notations for representing roman numbers etc.
All the notations which are used for a set (which is known as a collection of some common elements, possessing similar properties) are termed as set notations. 

A set can be written either in roaster form, which is also known as tabular form, or in set builder form.

For example: consider the following set:

Set Notation Examples
This is a set builder form which means, x belongs to Real number such that, x is greater than or equal to 3, and the same set can be written in roaster form (tabular form) as {3, 4, 5 ….}. This means this set is a collection of all numbers starting form 3 till infinity.

Consider the following diagram:

Set Notation Example
Here, to denote a set of numbers, we use capital letter of English alphabets, like; C is the set of complex numbers which includes all the Real numbers (R) which contains rational numbers (Q) and Irrational numbers (I), and all the other notations which represent the different sets of numbers, with their set notations examples are as follows:

Notation Meaning of the symbol  Example 
 { }  Symbol used for a set of collection of elements   A = {1, 2, 3}
 B = {3, 4, 5, 6} 
 $\mathbb{N}_{o}$  Symbol for the set of natural numbers including whole numbers starting with zero, $\mathbb{N}_{o}$ = {0, 1, 2, ...}
 $0\in \mathbb{N}_{o}$
 $\mathbb{N}_{1}$  Symbol for the set of natural numbers not including zero, $\mathbb{N}_{1}$ = {1, 2, ...}  $5\in \mathbb{N}_{1}$
 $\mathbb{Z}$  Symbol for the set of the integers $\mathbb{Z}$ =   {..., -2, -1, 0, 1, 2, ....}
 $-5\in \mathbb{Z}$
 $\mathbb{Q}$  Symbol for the set of rational numbers, $\mathbb{Q} = {x:x=\frac{a}{b};a,b\in \mathbb{N}}$  $\frac{5}{7}$$\in \mathbb{Q}$
 $\mathbb{R}$  Symbol for the set of real numbers, $\mathbb{R} = { x:-\infty<x<\infty}$  $5.5\in \mathbb{R}$
 $\mathbb{C}$  Symbol for the set of complex numbers, $\mathbb{C}={z:z=a+bi, -\infty <a<\infty , -\infty <b<\infty }$
 $6+2i\in \mathbb{C}$
 $\mathbb{Z}^+$  Symbol for the set of all positive integers, $\mathbb{Z}^+$ = {1, 2, ...}
 $15\in \mathbb{Z}^+$
 $\mathbb{Z}^-$  Symbol for the set of all negative integers, $\mathbb{Z}^-$ = {-1, -2, ...}  $-5\in \mathbb{Z}^-$
In the field of mathematics, whichever symbol or sign we use for calculations is termed as mathematical notation. Hence, all the notations stated above, whether statistical, geometrical, algebraic, calculus or any other numerals, all come under the category of mathematical notations.

Consider the following example:
Mathematical Notation
Here, x can take the value zero as the notation $\geq $ is given, but x can not take the value 4, because the notation is

Consider another example, by looking at the following number line:
Mathematical Notations
Now, we can write this information represented in number line, in the form of interval notation as [2, $\infty $], which represents:
x is greater than or equal to 2, which in mathematical notation is written as x $\geq $ 2.

The following is the list of some of the important mathematical notations used in various fields of mathematics, along with few examples:
Notation  Meaning of the symbol Example 
P(A)  Symbol for probability function, denoting the probability of event A  P(A) = 0.2
$P(A\cap B)$ Symbol for intersection, denoting probability of events A and B $P(A\cap B)$ = 0.2 
$P(A\cup  B)$ Symbol for union, denoting probability of event A or B $P(A\cup  B)$ = 0.2
$P(A\setminus B)$  Symbol for conditional probability, denoting probability of events A given B already occurred. $PP(A\setminus B)$ = 0.2 
$\parallel $  Symbol for denoting parallel lines  AB $\parallel $ CD
$\cong $  Symbol for denoting “congruent to”  $\Delta ABC \cong \Delta PQR$ 
$\sim$  Symbol for denoting similarity  $\Delta ABC \sim  \Delta PQR$ 
$\left | x-y \right |$  Symbol denoting distance between two points  $\left | x-y \right |$ = 5
rad  Symbol for radians  $180^{\circ}=\pi rad$ 
$\Delta $  Symbol for delta representing change  $\Delta = t_{1}-t_{0}$ 
$\Delta $  Symbol for discriminant also
$\Delta $ = b2 - 4ac
e  Symbol for Euler's number, e = 2.718281828.. $e=lim(1+\frac{1}{x})^{x}, x\rightarrow \infty $.
$\epsilon $  Symbol for epsilon denoting a very small quantity approaching to zero 
$\epsilon \rightarrow 0$
$\Pi $  Symbol for product of all values in series $\Pi $ xi = x1 x2 ....xn  
y'  Symbol for derivative, which is Leibniz's notation  (2x3)' = 6x2 
y''  Symbol for second derivative, which is a derivative of the first derivative again.  (2x3)'' = 12x 
y(n)  Symbol for the nth derivative  (2x3)''' = 12
i  Symbol for imaginary number i = $\sqrt{-1}$  Z = 3 + 2i 
 z* Symbol for complex conjugate of a complex number  $z = a+bi\rightarrow z^{*}=a-bi$
 $\theta $ Symbol for theta, which usually denotes an angle  $\theta = 90^{\circ}$

The above mentioned list of mathematical notations is just a small collection of notations from the vast collection of mathematical notations.
Algebraic notations are those notations which are used in calculating some algebraic equations, and hence, they are termed as algebraic notations. These notations are the part of mathematical notations only.

For example, consider the following notation:
Algebraic Notation
Here, the symbol used is Sigma, that is used for the representation of a total variable k starting from 1 to 5. Now, starting from k = 1, we will end at k = 5, and then add all the terms of the series, hence, the answer would be 15.

The other important rule for solving algebraic equations uses the rule of PEMDAS, in which the order of the notations, out of which notation have to be performed first, is very important for the accurate result, which can be understood by looking at the following diagram:

Algebraic Notations

After understanding about the order in which we have to evaluate an algebraic equation following the PEMDAS rule, we can now look at the following list of some of the important algebraic notations:

Notation Measure of the symbol, which represents the procedure
 +  Symbol for addition, known as plus  5 + 5 = 10
 -  Symbol for subtraction, known as minus  5 - 5 = 0
 X or *  Symbol for multiplication  5 * 5 = 25
 Ã· or /  Symbol for division  5 ÷ 5 = 1
 =  Symbol for equality, known as equal to  5 = 5
$\left |  \right |$  Symbol for absolute value
4$\left | 5 \right |$ 
 $\neq $  Symbol for not equal to  $4\neq 5$
 ( )  Open Parenthesis which excludes the lower and upper limit.  (2, 5)
 [ ]  Closed parenthesis which includes the lower and upper limit.  [2, 5]
 %  Symbol for percentage from 100  5% =  $\frac{5}{100}$
 $\sum $  Symbol for summation, used for addition of a series  $\sum x_{i}=x_{1}+x_{2}+...+x_{n}$
 <  Symbol for less than  4 < 5
 >  Symbol for greater than  6 > 5
 $\leq and\geq $  Symbol for greater than and less then equal to  x + 5 $\geq 6$
 x $\leq 1$
 f(x)  Symbol for s function of x   f(x) = 6x + 2
 $\pi $  Symbol for Pi, which is always a constant  $\pi = 3.141592654...$
 $\approx $  Symbol for approximate value  $4.99\approx 5$
 $\therefore  $  Symbol for therefore  $\therefore  $ 5 = 5
 $\infty $  Symbol for infinity  1, 2, 3, 4, ....$\infty $
 $\sqrt{}$  Symbol for square root  $\sqrt{4}=2$
 â‡’  Symbol for this implies   5x = 10 ⇒ x = 2
 â‡”   Symbol for equivalent   6 â‡” 6
 $\forall $  Symbol for “for all”  $\theta = 90^{\circ}$$\forall$ right angles
 âˆƒ   Symbol for there exists   âˆƒ x = 2 such that 5x = 10
 $\therefore$  Symbol for therefore  5x = 10 âˆ´ x = 2

Logical notations are those notations, where we have to decide or take a decision between true or false or both. As the name says, the concept of logic means applying a decision whether it is true or false. 

The following is the list of some logical notations which are very useful for taking logical decisions:

Notation Meaning of the symbol  Example 
 . Symbol used for "and"  a.b 
 ^ Another symbol used for “ and ”   a  ^  b
 & Another symbol used for “ and ”  a&b 
 + Symbol used for “ or ” a+b 
 V Another symbol used for “ or ”  a V b
 l Another symbol used for “ or ”   a l b
 x' Symbol for not - negation
¬  Another symbol for not negation  Â¬ a
 ! Another symbol for not negation  ! a
⊕  Symbol for exclusive  x ⊕ y 
Symbol for negation  ~ x 

Similarly, we can also convert the basic set theory notations into the logical notations as shown below: 

Logical Notation
Hence, logical notations can also be solved by using the notations of sets, to simply any logic. If we denote some statements with the letters as p, q, r or a, b, c etc, then we can make a combined statement by using the above listed logical notations. 
Notation which is specifically used to represent exponents is known as exponent notation. Exponent notation can be understood in a better way by looking at the following diagram:
Exponent Notation
Thus, we found that to use any exponent notation, we should be very clear about its base and its exponent. As shown in the above figure, 3 is the exponent of 2, which is the base and exponent is always written as the power of the base. This means 2 is multiplied 3 times, as 2 x 2 x 2 = 8. Some of the other examples of exponent notations are as follows:

Exponent Notations
Because exponents are used as a power to its base, exponent notations are also termed as scientific notations, where, a very large or huge number is represented as some multiples of 10. 

Only multiplies of 10, is allowed in case of scientific notations.

For example, look at the following number:

Exponential Notation

This number is very huge. So, to make it short, we will have to use a notation, which is the exponent or scientific notation. To convert this number, the first step would be to count the number of decimal places up to which we want its conversion. In the above example, we have converted this number into scientific notation up to 7 decimal places, by just counting from right to left the number of 7 places and then putting a decimal point after 7 places.
After putting a decimal, we will have to raise the power of 10 up to which we have counted, in this case it was 7.

Some of the other examples of exponent notations or scientific notations are as follows:
Exponential Notations