Influenza
The influenza virus is a very interesting system to work with from the standpoint of basic evolutionary biology. The virus evolves extremely quickly, on the order of 106 times faster than humans. This means that 10 years of virus evolution is roughly equivalent to 1 million years of human evolution. A lot is happening that we can directly measure, rather than being forced to make the sorts of inferences we are with other organisms.
Additionally, there are all sorts of interesting complications.
The virus genome is composed of multiple segments (think chromosomes) each of which harbors one or more genes. These segments reassort with one another, causing the evolutionary history of each segment to differ. Reassortment combines with influenza's ability to infect a variety of host organisms to facilitate movement of genes from one host organism to another. Hence, 'avian' flu and 'swine' flu. The human immune system primarily reacts to the hemagglutinin (HA) gene of the virus. This gene continually evolves in the human population allowing the virus to partially escape human immunity. However, an HA gene that has existed solely in birds for a significant amount of time will have not been vetted by the immune systems of the human population. Thus, if an HA gene moves from birds into humans, a pandemic is possible. Counteracting the effect of uniqueness in HA, other proteins in the influenza virus have adapted to their particular host environment. A strictly bird virus will not replicate well in the human host. This is where reassortment comes in. In rare circumstances, a mutant virus will emerge that has the HA gene of birds but other genes adapted to the human host. This can be bad.
The influenza virus is highly seasonal; most infections in temperate regions occur in the winter. This poses the question: "do chain of infections die out at the end of the flu season, or do they migrate southwards and continue their lineage?" I'm currently working on just this question.
Molecular evolutionary clocks
Generally, protein evolution occurs slowly at a rate of about 1 change every million years or so. However, each protein evolve at its own very specific pace with little deviation over millions of years. This by itself is rather amazing. I don't think anyone would have predicted this coming into things. Because of this, protein sequences have been described as "molecular evolutionary clocks." By comparing sequences from multiple organisms the amount of time separating these organisms can be computed.
By itself, the clock-like rate of molecular evolution suggests that the functional constraints imposed on a protein vary little over vast timescales. Meanwhile the organism in which the proteins function and the environment in which the organism resides change drastically. This in turn implies that cellular systems efficiently buffer proteins from changes in their external environment.
Although proteins generally evolve at a fairly constant pace, there are deviations. My own research focused on the form these deviations take. How clustered are the ticks of these molecular evolutionary clocks? What is the timescale of rate variation?