To my mind, some of the most interesting recent research in political science involves the use of geography. A map can obviously be very useful in conveying political information. For example, in a study I'm working on with John Sides and Lynn Vavreck on the 2012 presidential campaign ground game, we plotted out the precise coordinates of the Obama and Romney campaigns' field offices throughout the United States. The map below of Cuyahoga County, Ohio (which contains Cleveland) communicates the Democratic ground game supremacy in a way that a simple list of numbers and addresses can't. (See here for more details on the study.)
But geographic information can do far more than simply convey a concept visually. In the study I mentioned above, we found that proximity to an Obama field office was correlated with a greater Democratic vote share, while proximity to a Romney field office apparently conveyed no such advantage for the Republican candidate. The implication is that the Obama offices were more effective than the Romney ones, and we couldn't have found this without resorting to some geographic methods.
In a 2008 article, Shanna Pearson-Merkowitz and John McTague used more intensive geographic analysis methods to examine the dispersion of partisan voters in a few states. Their analysis showed that in some states, unsurprisingly, Republican voters are dispersed in a way that Democrats are not. For example, in the map below, we can see that there are several major centers for Republican voters in Florida (second map), while Democrats are more concentrated in the southeastern part of the state (first map). What is particularly interesting about this is that a more dispersed population of partisans is associated with a more factionalized party. As the authors argue, candidates can emerge from a centralized farm team and have enough support for a nomination. In a geographically dispersed party, however, rival factions will often emerge, requiring a divisive primary to determine the nominee.
Kristen Coopie Allen recently demonstrated the value of tracking the geography of campaign donations. She actually geocoded every donation to every state legislative candidate in the U.S. during the 2010 election cycle (nearly a million such donations) and looked to see how the geographic dispersion of donations was related to candidate success. What she found was interesting: Candidates with more dispersed donor pools tended to do better electorally, but those with a higher percentage of their donors coming from within the district tended to do slightly worse. The results suggest that just relying on your prospective constituents for donations isn't enough—you need outside money to prevail.
There are plenty of other fruitful uses for geographic information in political analysis. Todd Makse and Anand Sokhey, for example, have charted the geographic dispersal of campaign lawn signs, looking at how people end up getting signs and what kind of effects they have on the voters who see them. James Monogan, David Konisky, and Neal Woods used maps of wind patterns to discover that states tend to locate their most polluting power plants on their downwind borders, effectively exporting environmental and health damage to neighboring states.
Of course, maps aren't just used for analysis; they can also be very effective tools for political mobilization. For example, Susan Schulten highlights this map that charted the geographic distribution of slavery in an attempt to galvanize Northern opposition to the South's political power in the 1850s. (And in general, if you want to understand how maps can be used for political purposes, be sure to follow Susan's blog.)
Geographers have been using geographic information systems (GIS) like ArcGIS for many years, of course, but political scientists and other social scientists are starting to import some of these methods into their own research. This is definitely a positive development, as it appears that political phenomena are determined not just by who people are and what they believe, but also by where they live.
Update: I should have mentioned in this post that there is an excellent journal, Political Geography, that is devoted to exploring this intersection between politics and geography. Apologies for the omission.