Alignment algorithms to compare DNA or amino acid sequences are widely used tools in molecular biology. The algorithms depend on the setting of various parameters, most notably gap penalties. The effect that such parameters have on the resulting alignments is still poorly understood. This paper begins by reviewing two recent advances in algorithms and probability that enable us to take a new approach to this question. The first tool we introduce is a newly developed method to delineate efficiently all optimal alignments arising under all choices of parameters. The second tool comprises insights into the statistical behavior of optimal alignment scores. From this we gain a better understanding of the dependence of alignments on parameters in general. We propose novel criteria to detect biologically good alignments and highlight some specific features about the interaction between similarity matrices and gap penalties. To illustrate our analysis we present a detailed study of the comparison of two immunoglobulin sequences.