Algebraic topology is widely applied nowadays, not in the field of mathematics but in the filed of science too, especially, physics, computer sciences and economics. Applied topology is also widely used for the topological and mathematical analysis of shapes, their sizes and very huge ndimensional sets of data.
As we know that in algebraic topology we have to assign algebraic invariants to its topological given spaces, therefore, in applied algebraic topology, we apply this method to generate different functions resulting in various theorems, which are then further applied for various research purposes.
Consider,
for example, the Euler characteristic, in which we assign all the integers to its topological spaces, which is a homotopy invariant. It is generally denoted by X.
For example,
look at the following images:
Name of Shape 
Image

$\chi $ (Euler characteristic) 
Sphere 

2 
Torus 

0 
Mobius Strip 

0 
The next concept in applied algebraic topology includes the meaning of Simplicial Complex, which is a group of all the simplex (convex hull of all the points of a figure) kept together in such a way that the intersection of any two chosen simplicies results into a completely new simplex.
Regarding the above definitions and concepts, we define a Triangulation of the space as follows:
1).Each topological space maximum up to 3 dimensions is always homeomorphic to a simplicial complex.
2).Similarly, we can extend this definition in the field of vector spaces also, in which we can relate a vector space, on a field, mapped to n simplicies, where the base of the vector space is a simplex.
This concept of algebraic topology is applied to find the number of holes in a figure, which is the concept of simplicial homology.
3).Algebraic Topology can be applied by the method of Point Clouds, in which we collect points form a n dimensional space, and then prepare a topological method for solving it. In this method, we study a point cloud data by calculating its Hn(X) and then applying the Cech complex or Rips complex, whichever is applicable.
4).Another method for applied algebraic topology can be used as the Persistent Homology, in which a filtration F of a given simplicial complex X is calculated by making a sequence of its sub complexes in such a way that X
_{0} is contained in X
_{1} and so on up to X
_{n} which should be equal to X itself. We usually use the homology by making barcodes which shows its Betti numbers of its filtration.