The first and most important political skill any neutral or power player needs to have is an understanding of the alliances and important hot buttons those alliances are built around.
Political relationships are sometimes built on real friendships, but often times they are arrangements of convenience. Because of that, you can't just rely on who eats lunch with whom (although that's something to notice, too!) when trying to sort out the landscape. Most organizations have a handful of hot button issues that are central points along which battle lines are drawn. The web of relationships around those hot buttons are what creates the political environment.
An example might help -- In one organization I was involved in, there was a significant disagreement over centralized control versus decentralization. The battle over this issue was being played out in the way the purchasing organization was structured. Some people's positions were fairly predictable -- divisional operations VP's didn't much like a centralized structure, as it made getting their jobs done more difficult. Senior corporate management found a centralized structure more appealing because it was theoretically cheaper. But there were a couple of key senior corporate players who felt the decentralized structure led to better accountability and was worth the price.
In my example, the new vice president of corporate procurement made a lot of mistaken assumptions about who was on each side of the issue. In doing so the VP alienated potential allies, and solidified opposition. It was a big part of what led to the VP's eventual departure from the company.
So how do you figure all this out? First identify the hot buttons. There won't be more than a dozen or so of them in most organizations, and they tend to be the items people will talk about when they are having casual conversations. As you go along, gather information on who seems to be on each side of the issue. Try to base this mostly on what they say and do, and not who they are friends with or what their position title is, although that information can be useful at times. Then look for patterns.
People that tend to be in lockstep on most or all of the hot button issues, are likely to be allied. This will help you identify political camps and their captains. Bear in mind that this isn't a static picture. People, particularly fringe players or those who don't find an issue threatening or compelling, may shift their positions and alliances.
And whatever you do, keep your trap shut. Expressing your opinions strongly before you know where others stand is dangerous -- you have no idea who you might be making an enemy of.