In general, the Congress has much less in the way of power and influence in foreign relations than the President does. Congress generally takes a back seat to the President when it comes to foreign policy. Congress does have at least three powers that give it some say in foreign relations, but these are not very strong in most cases.
The first power that the Congress has is to declare war. This is not a very important power, particularly today. The reason that it is not very strong is that Congress has not been asked to declare war since 1941. All of the conflicts that the US has been involved with since then have happened without declarations of war.
The second power that Congress (more accurately, the Senate) has is the power to ratify or to reject treaties. This power is important in that no treaty can become law without the Senate’s consent. Of course, this power is still less important than the President’s power because Congress has no legal role to play in making the treaties. It can only vote them up or down.
Finally, Congress has power over all government spending. Congress can use this power to shut down a war that it does not like or to stop the US from aiding certain countries or groups. This is a very strong power, but it is hard to use in practice because it is often very difficult for Congress to oppose a war without seeming to oppose American military personnel. This means that it is very politically difficult for Congress to use this power.
All in all, then, Congress has much less power in foreign affairs than the executive branch has.