Yes, I love watching these!
I generally use them for games I don't own, have little expectation of an ability to play myself, or for genres that I'm not very good at (I love RTS and turn-based strategy games, for instance, but I'm terrible at them so watching someone who's actually competent play their way through Command & Conquer or Gemfire is a treat). For games I am good at or just enjoy playing, I will occasionally watch speed runs just to see how badly some games can be broken without resorting to save states and cheat devices.
Actually a little terminology may be helpful.
A "long play" (or longplay) is generally a start-to-finish run of a particular game with no commentary. The World of Longplays YouTube channel is an example of this. Exceptionally long games like RPGs are often broken up into multiple multi-hour videos, and occasionally they will be edited for the purpose of slicing out random battles with enemies that have already been fought or 'grind' segments where a player is just powering up to be able to take on the next boss, but otherwise they are the full game experience.
A "let's play" is also a start-to-finish run of a game, but with either the gamer or someone else providing voiced or text-based commentary and/or reaction. Kikoskia's YouTube channel is an example of this. There are two categories of Let's Plays, normal and Blind, and each are popular for their own reasons. Normal let's plays are usually done by people who already have experience with the game: they've played through it once, they know where to go, and they're like tour guides leading us through a specific title. The upside is there's little down-time in a normal let's play since the player is practiced and knowledgeable about where to go, and the commentator often telegraphs and foreshadows things so the viewer knows to pay attention to them. They're rarely caught by surprise unless the game itself is built on random elements, like Dwarf Fortress, Dungeon Hack, and so on. Blind, on the other hand, means the person playing the game is experiencing it for the first time; these are especially popular with horror games, because half the fun is seeing the player shit himself after a horrible experience. The downside of a Blind let's play is that sometimes the players get stuck on a puzzle, backtrack because they missed an item, or simply don't know what the next step is. It's doubly frustrating when you can tell what they're doing wrong, but you have to sit there for twenty minutes before they realize where they missed a certain key or whatever. I much prefer normal Let's Plays for games I've already experienced, and because the player's knowledge makes them more entertaining. Truly fun Let's Players make you believe they're actually in the shoes of the character they're portraying, something Kikoskia's quite good at in, for instance, his Doom 3 Let's Play.
A "speed run" is an attempt to complete the game as fast as possible, sometimes with conditions imposed to make it more interesting, but often it's a pure race from the title screen to the closing credits. Usually these runs are the result of people playing games to death and a community surrounding them coming up with new strategies that shave time off the current record results. I like watching Speed Runs of games I'm familiar with, because knowing the mechanics or what should normally happen makes it easier to understand just how (and how badly) the runner is breaking the game. There's a sub-category of speed run called the "TAS", or Tool-Assisted Speedrun, which uses emulators to play the game literally frame-by-frame to control the timing of button inputs and random number generation to produce optimal results that would be either extraordinarily difficult or impossible for a human player to pull off. TAS runs are amusing because the TAS player/programmer can showcase inhuman reflexes, show off hit detection boxes, earn critical strikes on every enemy, fire weapons so optimally as to never waste a single shot, and similar stunts (as showcased by this absurd TAS for Gradius on the NES).
I often will have one of these open in a different window when I'm chatting with friends online, just like some people leave a TV running in the background while they're doing other things.