Wow, most of you people don't know anything about IP law. Here are some answers that assume you're in the US:
Modding consoles is LEGAL, whether it is your own or someone else's you are modding for a fee. Installing a modchip WILL void your warranty, but is not a crime. Installing unlicensed software (ie, homebrew) can also void your warranty, though if you can restore your system to a pristine state (via a NAND dump or just uninstalling everything) it's unlikely that Nintendo will find out or care. It is also perfectly legal for Nintendo to revoke your ability to use their services (eg, repair services, Wii Shop, online multiplayer servers, etc) if they detect any modifications, hardware or software. If you are doing this for money, you should make sure your customers are aware of all of this.
Caveat to the above: under the DMCA, if you mod your console for the purposes of bypassing copy protection schemes, that is ILLEGAL. However, this is murky because the same modifications that allow you to illegally play pirated discs also allow you to run legal software. If you're not interested in piracy and your customers don't specifically mention it, then you probably don't have to worry about this.
Running homebrew is LEGAL. Selling homebrew is LEGAL as long as you abide by the license (as mentioned above, if the software is GPL licensed you must follow the GPL's requirement to make the modified source available OR link to the original source). However, this is very much frowned upon by the community, so you shouldn't do it anyway. If you wrote the software from scratch, or only using code with unrestrictive licenses, you are free to sell it.
Making backups of games you own is LEGAL. As part of fair use copyright law you are allowed to make one backup for archival purposes. You cannot sell this copy to anyone, you cannot even lend it to someone. If you ever sell the original copy you must destroy the backup. If the original is destroyed or stolen, you may continue to use the backup as if it were the original. It's not clear how the DMCA's restriction against bypassing copyright protection influences this, but from what I've read fair use rights trump the DMCA.
Making "backups" of games you don't own, or rented, or borrowed from a friend is ILLEGAL. That's piracy. It doesn't matter if you only downloaded a game to see if it's worth buying and delete it afterward. The only reason people around here call all disc copies "backups" is because real backups are legal while pirated copies are illegal.
Emulators are LEGAL. However, in order to use them you need ROMs, which are covered by the same rules as game backups. If you own a game, you are allowed to make a copy of THAT GAME (meaning, no, you cannot just go download the same file from the internet even if you have the disc/cart/whatever in your hand) and use that copy with your emulator. Same rules apply: only one copy, no sharing, you must destroy the copy if you sell the original game.