Topological Data Analysis (TDA) is a growing field in applied mathematics. TDA encompasses a wide variety of topological methods which can be applied to the problem of analyzing large or noisy point-cloud data sets. One of the most important of these methods is persistent homology, which characterizes the shape of a data set by finding holes of various dimensions in the data. This paper covers the algebraic construction of simplicial homology groups, gives efficient methods for computing them, and provides some intuition into what these groups mean. It then exnteds these ideas to a data analysis setting by discussing the algebraic theory behind persistent homology and also giving some examples computed in the R package TDA, which efficiently computes the persistent homology of data.