An emulator is a computer program that allows someone to play a game. These programs aren't exclusive to computers, as some games, like Animal Crossing, have an emulator within the game's programming to allow the player to play older games. However, for the most part, gamers use emulators on their computers to play classic or otherwise unavailable games.
How LPers use emulators
Some Let's Players use emulators to make their LPs when they feel it would otherwise be impossible to do that game. Most of the time, the games that LPers use emulators for are Game Boy and DS games, usually Pokémon though bigger LPers, like Chuggaaconroy, have used emulators to play games in the Mother series.
The legality of emulators is questionable. An emulator by itself is completely legal as it is just another type of software. However, the ROM's and ISO's which are used in the emulators are considered a form of piracy, when they are illegally copied from authentic video game cartridges and distributed online, therefore violating the original gaming publisher's copyright. For this reason, many LPers prefer using a capture card alongside the online game database on their game system of choice.