since repairing permissions is explained elsewhere and can be found easily enough with a search or looking at the stickies....
a beachball is one of two things, both mentioned already...
1. a beach toy that is inflatable and round and multicolored
2. when your Mac is 'busy' with an application and your mouse changes to a spinning multicolored ball that resembles a beachball. this only happens when a app is locked up momentarily or is having an issue it cannot resolve. usually more ram or force quitting the app helps
Beach ball of doom looks simular to the one below.
Repaireing premissions is a feature of OSX, accessed by programs. The routeens are built into OSX, but accessing them can be done by using a program us using the Terminal interface.
Run a program like Onyx or MacSweeper, or the Disk Utility. Have the programs update the prebindings, and if you can, run all the maintenance functions that they can. The former two are good tools (really just AppleScripts running routeens already built into OSX), I prefer MacSweeper