Which sink trap should I buy?
I'd just put a large pliers (channel lock) on that nut and snug it up. Clockwise from the top, assuming that the threads for that fitting are on the U-bend segment. Support the rest of the plumbing as you do so to prevent damage.

If that doesn't work, it looks like you have standard 1-1/2" (i.d.) plumbing there. Virtually any parts with the same configuration should be interchangeable.