The word подать (in the sense you're talking about) means "to pass something which is close, but that you can't quite reach".
You would say подай мне, пожалуйста, соль "pass me the salt, please" if you wanted someone to pass you a salt shaker which you could otherwise get yourself, but would have to stand up and bend over the table to do that.
Or, if you were standing on a step stool and trying to fix a lamp, you would say to your wife подай мне отвёртку "hand me the screwdriver" if the screwdriver were lying on the floor next to the stool.
Подать can usually be replaced by дать, but sometimes подать is better if you need to be more precise.
For instance, this phrase:
Он стреляет, а она ему подаёт патроны.
unequivocally means that he's doing the shooting, and she's standing next to him passing him the ammunition.
If you replaced it with давать:
Он стреляет, а она ему даёт патроны
this could mean that he doesn't have his own ammo, and she provides him with it — not necessarily while he's actually shooting.