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. The slightly slightly larger spacecraft used in Deep Space Nine are of a different, larger class, called runabouts -- see Danube Class Runabout for specifics. There were also tiny 'worker bee' variants called shuttlepods (a name also used for the NX-01 Enterprise's shuttles).
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, moreso than repeatedly destroying the series' main vessel, such as USS Enterprise, USS Defiant, or USS Voyager.
The Original Series
The first recorded victim (by our own timeline) was NCC-1701/7, Galileo, in the episode The Galileo Seven. It was later replaced by Galileo II. Kirk's Enterprise also had shuttles Columbus (possibly NCC-1701/2 ?) and Copernicus which seemed to survive the Original Series intact. Probably this was more due to limited special effects budgets which precluded their use than due to any special care by the Kirk Unit.
The Next Generation
A number of shuttles, including the captain's yacht Calypso, were presumably lost when Enterprise-D herself was destroyed. Before then several brave shuttles met their dramatic doom on the series, including:
The Voyager crew had shuttles of several different classes. They were quite careless with their shuttlecraft, but seemed to have the ability to replicate new ones during the mission.
Shuttles destroyed during Voyager's Delta quadrant sojourn include:
Plus the possibility of numerous other unnamed shuttle disasters, usually courtesy of piloting by Tuvok, Tom Paris, or Commander Chakotay. Often the fate of an abandoned or damaged shuttle was not made clear in the show, and fans usually charitably assume salvage where at all possible. One may assume that Engineering at least salvaged the warp core and computer core, as these would be problematic to replicate. I imagine there was a standard retrieval protocol for the cargo transporter during shuttle missions:
B'Elanna: Oh, crap. Transporter chief, Chakotay maneuver, now!
Eventually the show's writers realized the magnitude of the shuttle crisis, leading character Seven of Nine to suggest a new superior shuttle design. Designed primarily by hotshot pilot Tom Paris, Voyager's 'super shuttle' the Delta Flyer incorporated Federation, Borg, and other Delta Quadrant technology. The Delta Flyer lasted from its mid-series creation until the final episode of the sixth season of the series, when it was lost to the Borg. It was replaced for season seven by the Delta Flyer II.
Sources include Janet's Star Trek Voyager Site at http://www.leola.fslife.co.uk/scraft_missions.htm and Neutral Zone's shuttle database at http://www.neutralzone.de/database/Federation/Shuttle/ as well as www.startrek.com.