SpaceX is an US based company which builds and launches rockets into Space.
The BFR is a one of the rockets SpaceX is in the process of designing. It is intended to launch large payloads into space which could reach the Moon, Mars and other celestial bodies. The rocket has undergone a number of name changes and is a two stage to orbit (TSTO) super heavy launch vehicle (more than 50 tonnes to orbit).
The Booster (or Super Heavy) is the first stage while the Starship is the second stage. Both stages are intended to be entirely re usable.
SpaceX' mission statement has evolved from:
"SpaceX was founded under the belief that a future where humanity is out exploring the stars is fundamentally more exciting than one where we are not. Today, SpaceX is actively developing the technologies to make this possible, with the ultimate goal of enabling human life on Mars."(1)
"SpaceX designs, manufactures and launches advanced rockets and spacecraft. The company was founded in 2002 to revolutionize space technology, with the ultimate goal of enabling people to live on other planets."