Okay, on the vast majority of drill presses the chuck is attached by a tapered shank ( often called a Morse taper) a drill press chuck is almost always a three jaw chuck identical to a lathe. Each jaw has it own tightening hole ( the one you put the key in), the correct method of fitting an item into the chuck is to tighter each jaw gradually and in order ( yep a lot of people think these holes are spaced around the chuck for convenience). Now do we bother to do that, no, so an old machinist trick is to run the drill and gently tap the drill bit until it runs true (and don't let the boss catch you).the alternative to this in to remove the chuck and replace it with a Morse taper collet holder, basically this is like a router collet in a drill. You can buy a collet holder and a whole set of collets, from 1/16" up to ? . Not cheap but we are talking precision here (0.001 max runout and 0.001 out of round). To help with the confusion you can also buy 'tapping collets' for cutting threads and in the larger sizes morse tapered drill bits. Sorry I have probably caused more confusion than ever. So try the old machinist trick first.