There is a nice photograph showing an exploded view of a bottom bracket here: http://www.flickr.com/photos/kevoh/4433257157/
On each side, the ball bearings sit in the cup, and nustle against the outer edge of the raised section of the spindle.
To avoid the ball bearings falling out all over the place, firstly take the lock ring off and undo the non-drive side cup a little - we want to get that initial tightness out. Now undo the non-drive side cup a little bit more and once you have got it moving grab hold of the spindle on the drive side so that its ball bearings are being held in the drive side cups. Now tilt your bike over (or lay on its side) with the non-drive side down, whilst still holding onto the other end of the spindle. Put a rag underneath the bottom bracket, to catch any ball bearings which may escape. Keep pulling that spindle against the drive side cup, as you remove the non-drive side cup. It will come out with most, if not all, of its ball bearings. If any are stuck on the spindle you can pick them off with a screwdriver.
Now you have two choices - you can either let go of the spindle - some ball bearings will fall out and others may stay in the drive side cup; winkle the remaining ball bearings out with a screwdriver. Alternatively, grab hold of the free end of the spindle and keep pushing it against the drive side cup whilst you undo the it, allowing you to remove the cup and spindle together without any ball bearings dropping out. However, the drive side cup is usually very tight and, if you don't have the right tool to undo it (and even if you do) it is best just to leave it in place. If you do decide to remove the drive side cup, the chances are that you will have an 'English' bottom bracket which will have a left-hand thread, so needs to be turned clockwise to remove.
This is a bit of a conservative way to take dismantle the bottom bracket. To be honest, the ball bearings do not usually travel far, since they usually still have grease on them. If you put a rag underneath you can just dismantle the bb with the bike standing vertically without losing any bearings.
Before you reassemble, check the cups, bearings and spindle for pitting or wear, and replace if necessary. If you removed the drive side cup, ensure that you really tighten it up. Put grease in the drive-side cup and pop half of the bearings in there. While you have grease on your hands, do the same with the other cup. Stick the spindle in and then hold onto it while you screw in the non-drive side cup. Adjust the cup so that the spindle is running freely but without any play; hold the cup in place there as you tighten it off with its lock ring.
As for reusing cotter pins, it is so long since I have used them that I can't remember what to do. I think that the general advice was to use new cotter pins each time, but I am sure that somebody else will be able to give a more informed answer on this.