A Stands for Automatic. In this mode the car will shift gears for you unless you pull the selector to a certain gear. In that case it will select between that gear or lower: so if you are in "A" mode with the selector on 2 the transmission will chose between gears 1 or 2. BUT if it is in "A" mode and in gear 3 your car will shift faster (between gears - even faster than M)
M Stands for Manual. In this mode the car only shifts gear when you move the selector. BUT (at least in my '93 325i) it won't shift into 4th (out of 4 gears in our auto trannies). But this can be usefull when you want to cruise in a gear and not shift... or whatever...
Personally I just use 3 and its the best of all worlds... But in the mornings I use A and during some sort of fun-ish driving I use M.
Have fun: and also, M will not overide your computer to over-rev the engine, which means you can't fuck up your engine with M mode.