Which branch of the US government has the power of Impeachment?

The House of Representatives

Under Article I, Section 2 of the Constitution, the House of Representatives has the sole power to impeach (bring charges against) federal officials.

The impeachment process is similar to a grand jury indictment in a criminal trial, in that the House doesn't vote on guilt or innocence, but on whether there is a legitimate reason to try an official on specific charges (called Articles of Impeachment). If a simple majority of the House votes for impeachment, the official proceeds to trial in the Senate.
The House of Representatives has the sole power to impeach federal officials. The House of Representatives has this power under Article 1, Section 2 of the Constitution.