In terms of DNA mutations, it's usually just pure chemistry. The mutagens article gives a brief rundown of some of the usual suspects, but to pick a few of the ones I'm familiar with:
- UV light can cause pyrimidine dimers, which clearly is a dimerization of Cytosines or Thymines, which can cause replication issues.
- Deaminases, which remove amine groups, is probably the coolest example. Deamination can occur naturally, turning Cytosine into Uracil, but is often done deliberately as part of bisulfite sequencing to determine methylation status.
- Alkylation can occur, in particular of Guanine, which can lead to a preference in binding for Thymine; subsequent replication will result in an Guanine to Adenine conversion.
Similarly, there are many base analogs that will fill in for a given nitrogenous base. These analogs often prefer to bind to an "incorrect" base as above (and in the figure below with 5-bromouracil) which will lead to specific errors.
On a more intentional basis, biotechnological tools such as zinc finger nucleases, TALENs, and the (oh-so-sexy) CRISPR system can be used to specifically alter genome sequences of choice. ZFNs are in particular being used as a gene therapy tool for just that reason.
Otherwise, viruses (and transposons) that insert into the genome can cause specific mutations or errors if they do so non-randomly. Figuring that out, though, can be tricky; most oncoviruses cause mutations through specific genes that allow uncontrolled growth and tumorigenesis, not in a site-directed manner. There is, however, a growing hypothesis that certain retroviruses which tend not to cause cancer, in particular HIV, may be inserting into sites with certain characterizations, such as chromatin level. That is far from known, though.
If you're looking for random causes, that's easy. Anything that disrupts DNA repair mechanisms such as p53 or Rb will do that. For chemicals, intercalators such as ethidium bromide and thalidomide insert in between DNA bases and prevent proper DNA repair.