The President of the United States was just accused of accepting bribes while in office. What power does Congress have to bring charges against him? What are the specific responsibilities of the House of Representatives and the Senate?

Both the House of Representatives and the Senate have the power to remove political officials from office, including the President of the United States. In order to do this, the House of Representatives must first bring formal charges against a politician. If these charges stand, the Senate will try the politician and carefully look at the charges and evidence. The entire process is called impeachment. Any politician can be impeached, but impeachment doesn’t necessarily mean a politician will be removed from office. Removal usually occurs, however, if the politician is found guilty.

I am not really getting what you mean here, but he can be impeached (just invade you didn't know, impeached means thrown out of office)