How can I repair a broken bathroom sink drain?
Parts

Those valves look like those commonly used in the USA. In the UK I'd go to a DIY store and buy a "basin waste" but there is probably a different name in the USA. Basin wastes all seem to be a standard size over here, but yours looks small, I'd remove it and take it to the store with me. 



Tools

Tools needed include a large spanner or strap wrench (though you can succeed without these) and some plumbers putty. However the last waste I bought came with a foam washer that was used instead of putty and worked well. This was good because working with putty can be messy and frustrating if you are not used to doing plumbing.

The main body of some basin-wastes have two parts that screw together, sometimes these are difficult to separate (because of liberal use of plumber's putty). Although there are proper tools for this, you can improvise by using a large screwdriver (or similar rod) through the top opening into the ports (i.e. holes) in the side of the inside of the waste and use this to unscrew the top section. Yours seems to be broken and so should come out without any effort.

Procedure

Unscrew the retaining ring under the basin and remove the upper parts.

clean up the basin thoroughly both top, bottom and the internal bits you can reach (gunge tends to accumulate there)

Make a ring of putty, place it around the hole, fit the new waste and tighten, clean away surplus putty and reattach plastic waste pipe.