Some of the strongest evidence for the common ancestry of all life forms comes from genetics. Genetics is the study of how changes are passed down from generation to generation through DNA.
The basic units of heredity, called genes, are spread out along strands of DNA called chromosomes. Chimpanzees and other great apes have 24 pairs of chromosomes, while humans have just 23. If humans share a common ancestor with other great apes, as evolutionary theory proposes, we would predict that, at some point in the distant past, two chromosomes fused together in an ancestor of modern humans.
To test this prediction, we need to know something about the structure of chromosomes. While each chromosome contains different genes, all of them have two common features. The first are telomeres. Telomeres are repetitive sequences at each end of the chromosome, which act kind of like the plastic bits on the end of your shoelace that prevent it from fraying. The second feature in common is the centromere, a region in the middle of the chromosome that is used during cell division.
It turns out that the DNA sequence of human Chromosome 2 is nearly identical to that of two chimp chromosomes laid end to end. Furthermore, the human chromosome has leftover telomere sequences in the middle, as well as an extra centromeric sequence. This is like finding a shoelace with two of the plastic bits laid down end to end right in the middle that no longer serve a purpose. These non-functioning sequences are exactly what you would expect if two chromosomes fused in the past, confirming the prediction of common ancestry.
There are many other features of the human genome that point to common ancestry. Humans still have a remnant of the gene required for producing the protein found in the yolk of bird and reptile eggs. Mutations accumulating since the distant past have disabled this gene such that it no longer functions to produce yolk protein. This gene remnant was precisely located by scientists who predicted where it would be based on the location known from the chicken genome. This only makes sense if humans and chickens shared a common ancestor long ago. We have thousands of other such non-functional genes and each one tells part of our story as a species.
It is very hard to explain these and many other genetic features apart from common ancestry. Of course this evidence raises some theological questions.