Well, if it installed fine, there's no "running". If you install Advanced Power, it just replaces your battery status menu applet, adding some extra info next to the icon that pops up.

If you just installed Advanced Power Monitor, without Advanced Power, then nothing will happen. Advanced Power Monitor runs in the background and gets battery data, etc. Advanced Power Monitor, in turn, takes that data and displays it in place of the replaced battery applet.

So, the question becomes, did you install Advanced Power (which will install Advanced Power Monitor as a dependency) or did you only install Advanced Power Monitor (which will not install Advanced Power with it, and thus will just run in the background doing nothing - though you can query it from the command line, I think)?

If you just installed Advanced Power Monitor, then install Advanced Power as well - and you should be able to see what it does.