Dolphin pose is an interesting variation on Downward Facing Dog. Like Down Dog, it is a mild inversion, but in some aspects, it is more challenging than Down Dog. It takes some strength in the shoulders, arms and abdominals to maintain this asana. It is a really great pose to prepare the body to practice more challenging inversions, such as forearm stand and headstand.
You will notice when practicing this pose the deep stretch of your hamstrings, calves and arches of your feet.
An aspect of this pose that is more alleviating to the body than Down Dog, is that it relieves pressure on the wrists. Wrists can tend to become sore or agitated when practicing Down Dog regularly so Dolphin can help to diminish that negative aspect of the Down Dog posture.
I generally practice Dolphin to increase strength in my shoulders, and for the deep stretch that it provides to my calves. I also tend to get an agitated left arm and wrist at times while practicing Down Dog so Dolphin pose helps to relieve that issue while providing me with generally the same (and even more) benefits as Down Dog.
People who have shoulder or neck issues should avoid practicing this pose. For those who have very tight hamstrings, you should bend your knees while practicing this pose to alleviate tension in that area.
Ekhart Yoga has a great tutorial on Dolphin pose that you can find here.