I have a PCB that I am prototyping at home and it is double sided.... When I drill on one face the other face peels up a little.
Does any one know how to avoid this problem?
Your drill is probably not sharp enough. It is hard to keep an ordinary HSS (High-Speed Steel) twist drill bit sharp when drilling fiberglass. That is why special carbide drill bits are typically used for drilling PCBs. Another possibility is that you might be trying to drill too fast, pushing your way through before the drill has a chance to actually CUT the copper on the far side.
You can find many sources for special carbide PC board drill bits on Ebay by searching for: carbide pcb drill bits
It is prudent to buy them by the dozen because carbide is quite brittle and unforgiving of abuse. And for the same reason, you should only use them in a drill press (or equivalent) Never with a hand-held tool.
Another sources of carbide drill bits is ... - http://www.goldmine-elec-products.com/products.asp?dept=1273
Do the same thing the board houses do, and that is add another board of same size to both sides of the board your drilling.
They normally drill 5 to 15 boards at a time with a 'anti-burr' board on the top and bottom. It prevents any burrs in the copper. You may need to transfer your drill guide pattern to the upper anti-burr board.
That's the way the PCB industry does it, except their machines know where to drill and route holes by having it programmed into them.
NOTE: The bits wearing out will simply undersize the hole at first, then smoke a bit as they get older. By themselves the wearing does not cause burring, but you don't want to burn the boards either with worn out drill bits.