I recommend that you read the description and reviews here:
http://wdwig.com/tours.htm
We did it in '98 and really enjoyed it. You spend 30-45 minutes in the water, standing up to about your waist, "interacting" with the dolphins---we took turns stroking and hugging our dolphin, then sending him (her? I can't remember which) out to fetch something and calling him back.
Before that, you (1) get a tour of the facility and watch some health-checking behaviors, (2) watch a research session (w/Q&A with a research assistant explaining things), (3) get some training in a conference room about dolphins and what to expect in your encounter, and (4) change into your wetsuit.
After the encounter, you change and meet back at the conference room for more Q&A, and to receive a t-shirt, a video of your encounter, and other stuff.