Pairwise alignment between two biological sequences, such as DNA, RNA, or proteins, is a classical and well studied bioinformatic problem. In fact, this latter problem can be found in many biological analyses such as those involving data coming from sequencing processes. In this work we will formalize the computational problem, and we will present the two most common variants, which are the global and the local versions. More precisely, we will describe the dynamic programming algorithm proposed by Needleman and Wunsch for finding the optimal global pairwise alignment between two strings with respect to a given scoring function, and also the algorithm of Smith and Waterman for the local version. Here, we will provide the details of both the algorithmic procedures and some examples showing their usage in reconstructing optimal alignments. Finally, we will also discuss some alternative scoring functions that are used in practice to deal with the pairwise alignment of biological sequences.