A let's play is basically a rifftrax, only for a video game. Sometimes (most times), this means a recording of the game as the Let's Player plays it and makes funny commentary on it. The point isn't to do a speed run or show off any particular skills, but just to go through the game like a normal player and make it entertaining for people to watch/listen to. The other variation of Let's Play involve screen shots of the games, rather than video recording, with text instead of audio. These were popular when the idea of Let's Plays first started up a few years ago, but I'm not sure if they're still that big a thing. And some people like to mix and mash both, or have video, but use subtitles instead of audio.
The kind of entertainment value in a Let's Play varies from game to game, player to player. Some people just like hearing other people get really pissed off at a game, thus creating a niche for LPers that do nothing but scream profanities for hours on end. Other people like witty commentary, so clever LPers make jokes around the situational humor in the games. Other LPers focus on technical aspects of the game, some are all about screaming like small children during horror games, etc. There's pretty much something for everyone (everyone who likes watching other people play video games, I mean).
I admit, I'm not the biggest Let's Play fan (though I've watched my fair share), so I know there are subtleties to the genre I'm missing, but that's about the gist of it. Entertaining commentary while someone else plays a video game.
Edit 6/26/17: I wrote this before finding Markiplier and Jacksepticeye. I heartily recommend those two.