Echolink works through any repeater, doesn't need to be D-STAR. Your radio doesn't need anything special to use Echolink, it's the repeater that has the hardware. It's a gateway to the Internet, which connects repeaters together via a TCP/IP link. This is similar to IRLP or Yaesu's WIRES. The individual users on the RF side don't do anything other than using DTMF tones if they wish to make connections.
D-STAR is a digital mode, so the modulated RF is different. This would be as opposed to AM, SSB, CW (Morse) or FM analog modes, P25, DMR, Yaesu's Fusion. D-STAR repeaters can be connected together like Echolink, but this functionality also exists with Fusion, P25 and DMR. So there's a lot of redundancy here without all that much interchangeability.
Personally I'm using DMR and sometimes IRLP. Echolink adds the ability to use a computer as one end of the communications. What's not clear to me is if the person on the computer needs to be a ham if the person on the RF side is. I believe they would be since the destination to/from is RF and the link just happens to be highly complex instead of a mic at the end of a cord.