I've been following the speedrun-"scene" for quite a while, and recently I've gotten more and more into watching longplays too.
Obviously, a speedrun is a playthrough where the objective is to complete the game in as little time as possible. I'm sure many of you have seen the insane Quake speedruns that are out there.
Check out: www.speeddemosarchive.com for many, many entertaining speedruns covering almost all platforms! And all downloadable in the quality of your choice.
A longplay is just a recorded playthrough of a game.
Some resources;
www.recordedamigagames.com - has recorded playthroughs of literally hundreds of amiga titles, and are adding other platforms now too.
YouTube-users that have lots of longplays:
"cubex55" - (almost) everything from rec. amiga games + many PC and other platform longplays.
"ADVRipper4" - longplays of adventure games exclusively.
"DerSchmu" - longplays of C64 games.
"Kikoskia" - longplays of various games (narrated).
Ye who like this kind of stuff, post in this thread! I shall update the resources section of this post whenever somebody else makes me aware of a useful site.
Edit: Blast, there's a typo in the thread title. How can you change it?

it cannot be changed.

I did a lot of Duke Nukem 3D speedrunning once, mainly just for shits and giggles.

Fallout and Baldur's Gate have two of the most impressive speedruns I've ever seen.

I think recorded gameplays can be a good tool to see some games that i can't play because i don't own the platform they're released for (like the PS3 or 360 games).
For example, i was quite curious about the source of all the praise being tossed at Halo, than i watched some longruns.....and i couldn't quite get where it came from all the same :D

My favorite speedrun is this person... Somethingryan? on YouTube. He only does Tomb Raider speedruns, and he's really good at it. I'll post a link when I find it, but you can find it for me if you're so inclined.
I also read an issue of EGM (when they still existed) from September 2007 --the issue with the Halo 3 feature-- and one of the interviews was with Andew Gardikis who could do a speed run of Super Mario Brothers in five minutes. Yes, FIVE MINUTES.

Yeah they're like 90 minutes aren't they? Shows what you can do if you know the game that well

And the confusion got even worse when I actually had a chance to play the PC version. Just awful.
Anyway, about speedruns: I enjoy watching them every now and then primarily because these guys are really good at finding bugs. And when they do, you usually get a short glimpse below the surface of the game.
An easy example (you could actually see it in the original if you looked closely, but few people noticed it): In Mario64 there are the endless stairs, which you can only pass after you have done something (don't remember what). In the speed run, they exploit the slide-jumping (which is slightly faster than normal movement) to pass the stairs without performing the task first. And while they do, you can see that Mario gets teleported back a few stairs every now and then in order to make the stairs appear endless.