An Opponent Noise is an item you get from defeating someone during WiFi battles, similar to getting a Mega card from a boss. After beating someone for the first time, if they have a Noise Change (excluding Rouge (Burai) Noise), after you are awarded your card data, you will then get an Opponent Card in a similar fashion to being awarded a Hertz at the end of a match or mystery data from the MegaMan Battle Network series.
You are only allowed eight, so once you return to the versus menu, you will be forced to overwrite one of your preexisting Opponent Cards or toss out your new one if you already have eight. Though you only get one the first win against them, they will still get one from you once they get a victory, and you can still get another Opponent card from them if you search for another opponent, get them again, and win. This means that it actually is legal to have 6 Opponent Cards of the same person attached at once.
To actually use the Opponent Card, return to the main menu and go to Items. You can find them under the Key Items by hitting right. The names of the Opponent Cards are labeled by the handle of the person you got it from. Once you select it, you will be prompted to place it in any brother spot. These will take up a spot and can be removed at any time. If removed, the Opponent Card will not be deleted and can be reapplied at any time as well. Only by overwriting it after getting a new one can you delete an Opponent Card.
There are no differences between an Opponent Card and a Brother Card except that Opponent Cards do not give Link Power, they do not update with Data Update, and they do not give you access to their profile. Once used in battle, you can either Multi-Noise will their Noise or activate their White Card roulette exactly like a Brother Card.