Is it possible to power a cordless drill from wall socket adapter?

The answer is simple: Don't. If you did, odds are that you'd spend lots of money on high-current step-down transformers or other kind of power supply and run the risk of destroying your cordless drill. Corded drills, new from Home Depot, start at thirty dollars and will have as good or better performance than a $150 cordless. You'd be time and money ahead by just buying one-- and not have the headaches or risk of electrocution.

It would be possible to build such and adapter. It would depend on the rating of your drills and how you want to plug in the power.

Either way a cordless drill has a DC motor so it will require a step down transformer and a full wave bridge rectifier. Probably also need a regulator and a few capacitors. You might be able to use the power transformer from a laptop or some other device. But the voltage must match the drill and the ampere requirement must be very similar (more you can fry the drill less you can fry the power supply.)

As for how you attach the power supply I would recommend building the interface out of an old battery pack so you don’t have to modify the drill. But it should be possible to add an auxiliary power jack to the drill as well.

It's possible but by no means straightforward. An irreversible change is shown here: http://www.instructables.com/id/Convert-a-battery-drill-to-wall-power/

I could have sworn that I saw plug adapters for cordless drills before but for the life of me I can't find them now. I suppose it would be difficult - batteries are high-current beast and fitting a power supply that could manage that sort of current into a manageable size on the drill might be difficult.