
All Digimon start as a Digi-Egg. They are "born" in the Primary Village where Elecmon looks after them. From there they Digivolve into their Baby/Fresh form when their egg gets rubbed.
When they're in Primary Village each baby comes with a crib but when the egg is hatched outside -like Poyomon- the crib is gone. Their only attack is Bubble Blow.
From the baby form they Digivolve to the In-Training form. They get a little stronger and bigger but their only attack is still Bubble Blow. At this stage experience is needed to Digivolve further unless there's a Digivice.
Rookie form is after In-Training. The Digimon develop attacks of their own and are a lot stronger. Most Digimon prefer this stage because they aren't too big but they are still able to defend themselves. In order to Digivolve again they either need experience or a D-Arc.
The Champion stage is reached by swiping the blue card through the D-Arc. They have new attacks and more strength. The Digimon are able to reach this stage when they are full of energy and they're partner is in danger. The problem with using the card is that there is no set way to De-Digivolve down to the Rookie form again.
The Ultimate stage can only be achieved with a D-Arc, a Digivolving card and Calumon must be present to give off energy or a LOT of experience. Using the D-Arc and card is called Matrix Digivolving. When the Digimon first reach that level with the crest they gain more strength and new attacks but it takes a lot of energy. Again they have problems De-Digivolving.
Mega is one of the strongest forms a Digimon can achieve. A Digimon can reach it through merging with their Tamer -I think.. haven't seen it in practice but I saw a gif for it- or if they are experienced enough through normal Digivolving.

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A D-Arc tracks a Digimon's power. When the bars are full the Digimon is able to Digivolve. It also has a place to swipe the cards to give the Digimon the extra boost of power. It also works as a Digi-Dex, which identifies Digimon and displays stats.
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Cards have stored energy in them that when swiped through a D-Arc release the energy to the Digimon in one form or another.
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