Crests enable Digimon to reach Ultimate level.

After defeating Devimon, the children met Jennai for the first time, and he told them to go and find their tags and crests. These crests, coupled with correct care of their Digimon, would react somehow to help their Digimon become stronger.

The tags were found in an underwater convenience store (^^;;), and served as necklaces that would react when the children's crests were nearby. Crests were generally just random blocks in the digital world that shrunk to fit into the tag when it got close enough. Each has a symbol to represent it.

Tai was the first to get his crest, which made him think he had to be really strong and save the day, because he was the only one with a crest. He behaved stupidly and his crest reacted badly, causing Greymon to mutate.

Joe found his crest next, then Mimi, then Izzy and Matt at the same time, then TK and finally Sora, after some difficulty. But none of the children could figure out how to make their crests work.

Eventually, it just happened naturally. They found that their crests reacted for the first time when the kid who owned it strongly demonstrated whatever their crest represented.

Tai's crest is of courage. It first reacted when he got out to fight.
Joe's crest is of reliability. It first reacted when he saved TK from drowning but went under himself.
Mimi's crest is of sincerity. It first reacted when she was wishing she could do something to help stop Myotismon's chaos.
Izzy's crest is of knowledge. It first reacted after he regained his lost curiousity.
Matt's crest is of friendship. It first reacted when he saw that Joe was a good friend after all.
TK's crest is of hope. I don't know when it first reacts, I haven't seen that far yet. (Please don't give me spoilers. ^^;;)
Sora's crest is of love. It first reacted when she realised her mother really loved her after all.
Kari's crest is of light. It first reacted after Wizardmon died.

There was quite a saga with Kari's crest. Myotismon had it, and made copies of it that would react to her presence, which he gave to his many minions who were all searching for her. Wizardmon managed to get the real crest and gave it to Kari.

Later, Matt and Tai's crests assist their Digimon in warp-digivolving. The digivice helps Agumon/Gabumon digivolve into Greymon/Garurumon, the crest helps Greymon/Garurumon to digivolve into Metalgreymon/Weregarurumon, and a special power helps them digivolve to their final forms.