A handful of kids are given magical (well, make that digital) partners and told to save the world. Welcome to the Digimon Universe.
Digimon (short for "Digital Monsters") are data-based creatures of all colors, forms, shapes, and sizes who reside in a place known, creatively, as the Digital World. Occasionally the Digital World needs saving (evil conquerers are always popping up in places like that), and that's where our stories are set.
Digimon Adventure tells the story of several children who mysteriously end up in the Digital World while at summer camp. They meet their own personal digimon partners, receive "digivices," and learn about "digivolving" (the proces that allows a digimon to advance to a higher level, though said level may only be temporary.) They spend the season defeating bad guys, later learning that each chosen child (or, yes, "Digidestined") has a "crest"--the symbolic embodiment of a certain trait. (Love, Courage, and Friednship are examples.) The crests allow the digimon to digivolve to even higher levels than before, and come in useful as thhe kids meet up with sucessively stronger and stronger villains. By the end of the season, the world is saved, summer is almost over, and the children and their digimon have a tearful parting. The youngest members of the group, however, seem to know theat this is not thhe last time they will see eachother.
Digimon 02 is set several years later. A group of five children (the two youngest from the previous season and three new members) are called to fight a new menace in the Digital World. This time, their enemy is human...and the digimon cannot digivolve. The Digimon Emperor (as he styles himself) is a middle-school kid about the age of the current Digidestined children, and has found ways to control otherwise "good" digimon, prevent normal Digivolution, and just generally be evil. Two special forms of Digivolution are presented this season: "Armor" forms are activated by "Digieggs" bearing the crests of children from the previous season. DNA-digivolving allows two digimon to join together into one creature. The Emperor eventually joins the "good" team, and all the children together face the ongoing threats of Digimon invading the real world. The season ends with the return of the last season's most formidable foe, and an epilogue to the lives of the children from the first two seasons.
Digimon Tamers is set in a different world than the first two seasons--one that is supposed to be our "real" world. The Digimon card game is very popular, and a few lucky children--for some strange reason--receive real digimon. The threats in this season range from a playful imp, to 12 mythical digimon known as Devas, to a human intelligence group set to exterminate Digimon altogether, to a soulless young girl with a demon-possessed sock puppet. The children are able to use the cards from the game to equip their Digimon with various abilities, and later they are able to join with their Digimon into one form.
Digimon Frontier, like Digimon Tamers, is a completely new continuity. In it, four new Digidestined are able to become digimon themselves.