D-3 Digivice

The D-3 Digivice is so named, because of it 3 capabilities: Digital, Detect, and Discover. By Digital, Detect, and Discover, it means:

Digital: Digivolution capabilities

Detect: Finding other digi-destined

Discover: Finding Digi-Eggs

how everybody got their D-3: 

Tai was being chased in the digital world by a digimon with a dark ring when he took shelter in a cave, and saw something in the back on a pedestal. When he tried to touch it, 3 beams of light came shooting out of the object, went through a digi-port, and came out of a computer at a school. The beams shot into the hands of Davis, Yoli, and Cody, where they became digivices.

Kari and TK's old Digivices became D-3s when they found digi-eggs in a cave that allowed Gatomon and Patamon to armor digivolve.

D-Terminal

The Digi-Eggs that the digi-destined found were kept in their D-Terminals. When the digimon needed to digivolve, the D-Terminal would produce the digi-egg of choice. 

Control spires, placed throughout the digital world by the digimon emperor, kept the digimon from digivolving, so they had to armor digivolve.