It's a grey area, and as such the general rule here is that if you're not sure, it might be better to avoid it.
The reason it's so complicated is because if you own a physical copy of the game that you paid for, there's really no reason you shouldn't be able to emulate a digital version. Playstation and Playstation 2 emulators will even boot the games right off of their physical discs from a normal PC DVD player. But a lot of companies liken it to piracy because they're often distributed the same way, there's really no safeguard to ensure that only people who own the game will do it and there's no way of verifying that the person actually owns the game anyway, so that becomes more of a personal moral justification than any legal claim.
On the other hand, especially for older games, most websites that host Let's Plays won't care if you're emulating or not and the major companies haven't really taken any steps toward stopping videos of emulated games like Pokemon, Sonic the Hedgehog, or Super Mario Bros. 3 from popping up on the Youtubez.
So in conclusion, I won't speak for its full legality one way or the other because I just don't really know, but from a moral standpoint if you're unsure, then don't do it; at the very least not if you don't own the games already, because then it almost assuredly would be piracy.