Starship is the name of the 2019 version of the second stage of the SpaceX super heavy lift vehicle.
Conceptually, Starship ia a Single Stage to Orbit (SSTO) Mars transportation vehicle, that can be launched from Earth using a first stage called the Booster, or Falcon Super Heavy. In that configuration it is a Two Stage To Orbit vehicle (TSTO)
It was presented by Elon Musk during the announcement of Yusaku Maezawa' Dear Moon project, as an evolution of the BFR/BFS concept and Interplanetary Transportation System (ITS) concepts.
Originally planned to be constructed of carbon fiber composite, it was changed to a Stainless Steel design in January 2019 .
85-120 tonnes mass, 9m diameter, 100-150 tonnes of payload to LEO, 100-150 tonnes to Mars. These are target values, the lower the mass of the vehicle, the higher the payload mass will be. Payload volume of 1000 m3.
3 vacuum Raptor engines with 380s ISP and 3 atmospheric raptor Engines with 330s ISP. Nominal thrust of 2000 kN, (200 tonnes of force per engine) These numbers are subject to change as the engine and the vehicle concepts are under development.
120-160 day transportation time to Mars, using aerocapture at Mars.
Fully reusable, rapid turnover and low maintenance vehicle.
Up to 100 passengers to Mars, although this has not been demonstrated yet by SpaceX.
The fundamental enabling technology of the Starship is supersonic retro propulsive landing on Mars. The use of supersonic retropropulsion in a critical phase of the Mars entry path allows the vehicle to land heavier payloads that previously thought possible. Although the exact details are not public, the current SpaceX Falcon 9 booster rocket has done flight tests that would confirm the flight path. 
A second enabling technology is the capacity of refueling in orbit.
A third enabling technology is the use of methane as fuel, than can be provided by In-situ ressources production systems on MArs, and therefore allow for the re-use of the spaceship.
A fourth technology is a robust heat shield for Mars and Earth entry. This allows for fast re-use and lower costs, but also for faster transit times, reducing the radiation exposure to travellers. The Spaceship is not intended to use low energy Hoffman transfer orbits, but higher velocity orbits. These have lower transit times but leave the vehicle with significant velocity when it reaches Mars or Earth. The Starship must then use direct entry and aerodynamic braking to shed the kinetic energy from the extra velocity.
The NASA Ames research center trajectory browser can be used to explore transit times to Mars and other bodies in the Solar System. Trajectory browser