Crossup is the act of crossing over to the other side of an opponent’s block (i.e. behind them) in hopes that they do not know the exact moment when to switch their blocking direction (left or right). Less commonly, it can refer to an attempt at this.
In 2D games, where you can generally jump over your opponents, and crossup is an important factor in offense in old-school and old-school-like games (i.e., it is slightly less so in games where preemptive blocking is important; it is not such a big deal in combo games to the point where some [e.g. Melty Blood] do not even allow the possibility). The bulk of crossup attempts are performed using jump attacks for two reasons:
In games where crossup is important, some moves will be better at crossup than others; an attack hitbox facing more forward than backward when going right over the opponent would usually be a better choice than one that points down neutrally.
Crossup can also be very effective when frontsteps or teleport moves can go behind an opponent. And any one of these methods (i.e. frontstep, teleport, or jump attack) can be effective when peforming meaties (depending on the game and the character, some approaches will be better, obviously; it would be ill-advised to try a jump-attack crossup against someone with an invincible anti-air move, for example!).
Some moves, such as Iori’s Yuri Ori, are quite obviously intended for crossup only (it attacks “behind” him...! But this is actually towards the opponent if he has already passed over).
The actual strategy of crossup originated from player research with Street Fighter II; it was not intended by the developers, but since it is a pretty fair and easy maneuver to understand, it has been incorporated in many, many games ever since.
In 3D games, you do not really jump over your opponent (definitely not the same way as in 2D, at any rate), but rather, you attempt to shift axes, as does your opponent.
It depends on the game as well, but depending on who has which axes, there come times when the opponent will be unable to block things from your direction (e.g., they have their back to you, and very much not on purpose), so it becomes very difficult for them to to defend themself. However, situations where you can get this are very rare against a competent player in a typical 3D fighter, so there isn’t much point in attempting to achieve it; you have to take them as they come.
Naturally, launching moves and such will allow you to get behind the opponent during juggles and such, but this is not really the same thing as crossup at all.