Booster (or Super Heavy) is the current name of the first stage for the SpaceX super heavy lift vehicle. The vehicle was announced during the 2019 Dear Moon project presentation by Elon Musk and Yusaku Maezawa.
Super Heavy is a 9m diameter rocket stage, 68m high, powered by 37 Raptor methane-oxygen engines intended to be used with the SpaceX Starship vehicle. The vehicle carries up to 3300 tonnes of propellant.
Super Heavy is intended to be recoverable, with rapid turnaround between launches.
Originally planned to return and land on the launch mound, the first version will have landing legs. Elon Musk has pointed out that 'the best part, is no part'.
The current plan for operating Starship Boosters, Tweeted by Elon Musk, it that they will be caught by a grabber attached to the landing tower and won't need landing legs at all. The high design iteration rate of SpaceX for the Starship means that many solutions may be tested and various solutions used depending on the situation.