There are different ways to launch your foes through the air, as well as different goals, depending on the game being played.
Character action games, such as Devil May Cry, God of War, or Bayonetta, allow you to launch enemies into the air for the purposes of either following them and performing an air-combo or remaining on the ground to keep them suspended, usually with gunfire. Some fighting games, like Marvel VS Capcom or Guilty Gear, have launch moves that allow for air combos.
At times, launching your foe into the air can be the goal of a game, such as the Super Smash Bros. series. The goal in that case is to do enough damage to your opponent that they can be launched to the extreme of any of the four screen directions.
In games with their own physics engine, such as Half-Life 2, Psi-Ops, or Mass Effect, you can use your character's abilities to launch enemies through the air and cause either falling or smashing damage. At times in games like these, the enemies themselves can be used as weapons by launching them at one another.
is the act of launching an enemy or opposing fighter into the air and trying to keep them there for a prolonged period of time, rendering their attacks and defense useless. A notable example of juggling can be using Dante's uppercut slash in Devil May Cry to launch an enemy into the air, and suspending them there with your pistols until they are eliminated.
In some fighting games, juggling can be particularly devastating, as it it usually catches the other fighter in a loop that is hard to escape. Some less-balanced fighting games unintentionally include infinite juggles. Based on the skill of the player, this juggle can go on until all of the other fighter's health is diminished.