Since the idea of space travel first began appearing in print, and later in film, it has taken shape with a naval metaphor: star-ships, captains, decks, bridges, and so forth. Traveling from island to island, you stay inside an intergalactic ship, sails (or hyperdrive) unfurled to the winds. When you arrive, you take a dinghy to the shore, or a lifeboat, or a skiff. A small craft, stripped down for the much-shorter voyage from the mothership to the shore, with only the bare essentials for a day trip. In the naval metaphor, the drop ship is the captain's yacht, the dinghy, the rowboat of interstellar travel. The "bare essentials" for a drop ship are life support, heat shielding, and propellant for the trip back up. Carrying the necessary propellant for the trip back up is what separates a drop ship from a reentry vehicle, SSTO transport, or a space shuttle.
I was unable to discover who coined the phrase "drop ship" to refer to the down-and-back-up craft that is ubiquitous in space science fiction. Robert A. Heinlein's novel Starship Troopers (1959) features a "lifeboat" which descends, picks up the troopers (minus their ammo) and returns to the Roger Young. The movie Aliens (1986) was the first movie to introduce the term into widespread sci-fi vernacular. Aliens deserves the credit for entrenching the idea of a drop ship in the minds of pop culture: for every factual reference to the idea of a drop ship on google, I found many references to the Space Marines and their "UD-4L Drop Ship" as depicted in the film. During the same time period (1985-1987), Battletech RPG sourcebooks were being published, extensively detailing drop ship operations. It's possible that both sets of authors were inspired by something slightly earlier; perhaps even Heinlein's lifeboats. All the same, the concept is nothing new: as early as 1928, covers of Amazing Stories showed stellar explorers exiting a "landing craft" and walking onto the lushly forested surface of one of Jupiter's moons.
The first drop ship ever made on Earth was the Apollo Project's Lunar Module, circa 1969. It descended from lunar orbit, landed safely, and allowed the crew to exit for an indefinite amount of time--limited only by their life support and sustenance needs--and then ascended from a standstill (leaving unnecessary hardware behind) to dock with the still-orbiting service module-command module complex. The space capsules for the Mercury Project (and likewise the Gemini Project, and any Russian equivalents) don't qualify because they were designed with an "up-then-back-down" mission, vice the "down-then-back-up" mission which makes drop ship design so challenging.
Don't get me wrong: single-stage-to-orbit designs are the most promising avenue of research for an eventual all-purpose drop ship. Scramjet technology, ablative heat shielding, regenerative cooling, and possibly nuclear fusion will all need to come a long way before we're there (current rocket designs don't have the impulse density necessary to do the job and leave room for cargo). Of course, by the time we develop actual interstellar travel, we'll have done all the "baby steps" like a mission to Mars, and we'll have learned some of the important lessons necessary to building a reliable drop ship.
Don't confuse the spacecraft called a "drop ship" with the method of package delivery called "drop shipping" where the retailer tells the wholesale source or factory to send a package directly to the customer's address. Drop shipping technology is already very reliable, and compared to the cost of a drop ship, unbelievably cheap. If it needs to get from the factory to the customer, drop ship it; if it needs to get from orbit to the surface and back, send it in a drop ship.