How can I mend a broken bathroom sink drain pipe?
What you need to replace is the "drain assembly" or "pop up". Here's a link to a sample:
http://www.homedepot.com/p/Delta-Push-Pop-Up-Drain-Assembly-in-Chrome-with-Overflow-Holes-72173/204154080

There's every chance you've got a 1-1/4", but maybe it's 1-1/2", so bring your old part with you to the plumbing place or big box.

Don't buy a plastic one. They're terrible.

I'm assuming you've not got a lever assembly to close the drain. (Typically, that's a little rod behind the faucet.) Just be sure to buy one that matches what you have now.

Beyond that, you'll need a big wrench to undo/redo the nut, and some plumber's putty to get back in business. (Some instructions these days suggest silicone around the top of the assembly... I wouldn't do it.)