Juggle is a term for using a combo to repeatedly hit a character in the air without letting them recover or hit the ground. The name comes from the fact that it vaguely resembles the motion objects make when juggled.
The most common way to juggle is using continuously Up Tilts, but Up aerials can also be used, especially for air juggling. Some characters, such as Kirby, can also "juggle" with their neutral ground attack. This is an effective technique to rack up the opponent's damage. But as damage increases, juggling becomes naturally harder, since the opponent starts flying farther, eventually putting an end to the juggle.
A character's vulnerability to juggling depends on their size and falling speed as well as the juggling situation. Naturally, the larger and heavier a character, the more vulnerable they are to juggling. Fast fallers tend to be more vulnerable to ground juggles such as Fox's u-tilt juggle since their fast falling speed keeps them in the u-tilts reach. Floaty characters tend to be more vulnerable to air juggles such as Meta knight's uair juggle since they take longer to fall back to the ground, giving them more time in the air which leaves them vulnerable to more air attacks. As such, fast fallers are less vulnerable to air juggles since their falling speed causes them to stay in the air less, making them more difficult to follow up aerial attacks on. Floaty characters are less vulnerable to ground juggles since their slow falling speed causes the u-tilt to push them away from the juggling character.
Some characters, like Mewtwo, have ways to escape juggles (Teleport, in its case). Air dodging is especially effective, as are momentum-changing aerials. Some characters have a Stall-Then-Fall move that can be used to counter juggling. Using a counterattack such as Marth or Ike's Counter works as well. Having fast down aerials with large hitboxes are also useful for stopping juggles. Kirby's Stone is another useful way to avoid juggling, as it gives him temporary invincibility while he is in his stone state. It also has very powerful vertical knockback, which can KO a careless juggler.