Emulators work slightly differently. The disclaimer states that you can download all the games you want but you can only keep them if you ACTUALLY own the game as a product (i.e you have the CD or cartridge in your possession). Other than that, you are allowed to download the games but you should delete them from your system within 24 hours. If you decide to keep them then it's on your own head...which is fair play I reckon.
Emulation sites claim this, but the law itself states otherwise. The whole 24 hour thing is complete bull, and owning the actual game still makes the creation or possession of a commercial ROM illegal. The disclaimers on emulation sites are mainly there to
1) Make the users feel better
2) Keep themselves out of trouble
Possession of something you didn't pay for isn't magically legal for the first 24 hours, no matter what it is. I don't know where this hooey came from. I, too, have seen this posted on countless emulation sites. It makes no legal sense whatsoever.
There is nothing legal about emulating commercial games, whether you own them or not. This was brought up in Nintendo Power one day some years back, and this is what they had to say about it. I'm not saying I'm against emulators. I'm just stating what I've heard. Personally, I see emulation as a way to keep the classics alive, and to allow users to get back into their favorite games at higher quality than they ever could on the original system (an N64-style game in 1600x1200 resolution looks pretty impressive, really

)
Otherwise it's like renting a game for free if you don't own it. Even if you delete it in 24 hours, it takes the business away from the video rental companies. Using a ROM of a game you own, I personally feel *should* be legal, and technically in Canada it is (as long as we don't sell them), but from what I've heard, in the USA, commercial game ROMs are Illegal, as the code is confidential property of the game producers.
Emulators themselves are Not illegal. It's the ROMs that are, whether you own the original game or not. I don't know the Exact legalities behind it, but I know I've heard at least this much.