Speed Demos Archive is an excellent site that I have been going to for a while. I mainly watch runs for older games that I used to play myself. Last summer I tried my hand at a speed run for the game Sonic Spinball but I never finished it because that game is hard as shit to run. Earlier this month I was thrilled to see that a run had been completed for it.
Three of my favorite speedruns I've seen:
Sonic 2 by Joe Stanski:
(Note: This video has his commentary.)
Super Mario 64 100% by siglemic:
Toe Jam & Earl by InsipidMuckyWater: