In the Star Trek universe, shuttlecraft (informally 'shuttles') are short-range craft carried aboard a starship or starbase. They are used for personnel and cargo transport in situations where transporters are not an option due to range, risk, or the need to relocate irascible doctors.
Runabouts are slightly larger versions of these spacecraft with more range, power (including warp drive), and weapons capacity than shuttles. They also boasted a short-range, two-sophont transporter and a modular mission-specific mid-section.
Runabouts were first introduced in the Deep Space Nine series. DS9's runabouts are all of the Danube class. USS Danube (NX-72003) was the original, prototype runabout - this craft was never seen in a Star Trek episode, but is mentioned in other Star Trek materials.
Deep Space Nine's runabouts are named after the rivers of Earth. DS9 started out its tour of duty as a Federation space station with three runabouts. These dropped off with the initial DS9 crew by USS Enterprise. The initial trio were the Yangtze Kiang, the Ganges, and the Rio Grande.
Destroyed Runabouts
Life is nasty, brutish and short for the shuttlecraft and runabouts of the Star Trek universe. A popular hobby among franchise enthusiasts is tracking the damage and destruction of these small ships inflicted by the writers of each series. Shuttles and runabouts are destroyed quite often - it's dramatic but (semi-)plausible, more so than repeatedly destroying the series' main vessel, such as DS9 or USS Defiant.
Among the fallen:
... and several others. Three cheers for the indestructible Rio Grande! (NCC-72452), the only original DS9 runabout to survive the series. Also surviving were Rubicon (the replacement for Mekong) and Volga.
Sources include Vessels of Starfleet at http://steve.pugh.net/fleet/danube.html and
http://www.startrek.com