That is an excellent question. People are indeed able to weld things underwater. Usually this involves ships or offshore machinery. The welding is usually accomplished through shielded metal arc welding (SMAW). Its a type of welding that uses an electrode which is sheathed in flux. The flux evaporates and creates a gas environment around the weld which shields the weld from the outside (wet) environment. The exact type of electrode material used will depend on the metals that you are trying to join, but generally they might be some type of steel alloy.

I hope that helps.