Hindu pushups involve going from the position (asana) Downward Facing Dog (
Adho Mukha Svanasana) to Upward Facing Dog (
Urdhva Mukha Svanasana) and back again. In my Vinyasa and Ashtanga yoga classes we usually don't go directly from one pose to the other, but instead do this sequence:
Downward Facing Dog ->
Plank -> Pushup (
Chaturanga Dandasana) -> Upward Facing Dog -> Downward Facing Dog
Hindu Pushups are much favored by bodyweight exercise proponents (most notably
Matt Furey) and ultimate fighters. They are excellent for the upper body and spinal flexibility. As for side deltoid benefits, I have no idea.