Evidence for evolution comes from many different areas of biology:
Anatomy. Species may share similar physical features because the feature was present in a common ancestor (homologous structures).
Molecular biology. DNA and the genetic code reflect the shared ancestry of life. DNA comparisons can show how related species are.
Biogeography. The global distribution of organisms and the unique features of island species reflect evolution and geological change.
Fossils. Fossils document the existence of now-extinct past species that are related to present-day species.
Direct observation. We can directly observe small-scale evolution in organisms with short lifecycles (e.g., pesticide-resistant insects).
Like structural homologies, similarities between biological molecules can reflect shared evolutionary ancestry. At the most basic level, all living organisms share:
The same genetic material (DNA)
The same, or highly similar, genetic codes
The same basic process of gene expression (transcription and translation)
The same molecular building blocks, such as amino acids
These shared features suggest that all living things are descended from a common ancestor, and that this ancestor had DNA as its genetic material, used the genetic code, and expressed its genes by transcription and translation. Present-day organisms all share these features because they were "inherited" from the ancestor (and because any big changes in this basic machinery would have broken the basic functionality of cells).
Although they're great for establishing the common origins of life, features like having DNA or carrying out transcription and translation are not so useful for figuring out how related particular organisms are. If we want to determine which organisms in a group are most closely related, we need to use different types of molecular features, such as the nucleotide sequences of genes.
Biologists often compare the sequences of related genes found in different species (often called homologous or orthologous genes) to figure out how those species are evolutionarily related to one another.
The basic idea behind this approach is that two species have the "same" gene because they inherited it from a common ancestor. For instance, humans, cows, chickens, and chimpanzees all have a gene that encodes the hormone insulin, because this gene was already present in their last common ancestor.
In general, the more DNA differences in homologous genes (or amino acid differences in the proteins they encode) between two species, the more distantly the species are related. For instance, human and chimpanzee insulin proteins are much more similar (about 98% identical) than human and chicken insulin proteins (about 64% identical), reflecting that humans and chimpanzees are more closely related.