The search algorithm is a step-by-step procedure to find specific data and retrieve elements from any data structure, where data structure plays an important role in raising the performance of a program or software. The software should store and retrieve the data needed as fast as possible. The hardest part for the programmers is to choose the appropriate data structure for a program, according to the type of operation is required. The breadth-first search and depth-first search are two examples of the search algorithm. Breadth-first search is the simplest traversing algorithms for searching a graph, it can handle a graph with billions of vertices and edges. One of the main problems with the breadth-first search is that it consumes a lot of time. While the depth-first search is an algorithm for traversing or searching tree or graph data structures. However, both of them are widely used for various data structure and have many interesting applications. Our goal is to give an overview of both algorithms, implementing them and describing some of their applications.