What powers do Congress and President share?

The president has veto power over bills passed by Congress.

The Senate has the power to confirm or reject nominations that the President makes for ambassadors, cabinet member and federal judges.

The Senate has to ratify any treaties the President makes by a 2/3 vote before they can take effect.

The president can send troops to some foreign country and essentially make war, but only Congress can declare war and appropriate the funding for military action.