Autonomous Spaceport Drone Ship is the name that the U.S. private company Space Exploration Technologies, a.k.a. SpaceX, have given to their oceangoing rocket retrieval barges. While ships have been intimately involved with spaceflight since the very beginning, mostly because oceans provide lovely enormous areas where it's generally okay to drop fast-moving metal objects when you're done with them (or when you're not entirely sure exactly where they'll come down and you really want them to land softly) this is a new chapter.
SpaceX has embarked on the mission of making access to space cheaper - cheap enough to make human-rated spaceflight more accessible and, if you believe Elon Musk, cheap enough to colonize Mars in his lifetime. This is a tall order. One of the reasons that spaceflight from Earth is so very expensive is called the Rocket Equation. In order to reach orbit, a spacecraft must both expend enormous amounts of energy as well as be ruthless about dropping dead weight as soon as possible. If rockets such as the Soyuz, Saturn V or SpaceX's own Falcon 9 were to keep the entirety of their structure for the entirety of their travel, they would likely never be able to reach orbit. Even if they could (I haven't run the math, but I doubt it'd be possible, see the Rocket Equation, above) they'd never be able to carry useful payload.
So, from the earliest days of modern spaceflight, rockets have used stages - as each section of the rocket (the first is generally called the booster) empties itself of fuel and oxidizer, it is detached and allowed to fall back to Earth. The remaining vehicle, freed of the killing mass of this now-useless section, can use fewer engines and less fuel to continue on its way.
This is one reason the U.S. situates its space launch facilities next to oceans. As mentioned above, these stages can be allowed to fall into the ocean with little risk, and traditionally have been. In some places, other space programs, without handy oceans, have allowed their hardware to drop onto (mostly) uninhabited areas. Despite how dangerous this sounds, I don't believe any humans have been injured by a falling booster stage. The Soviet Union and Russian space program, launching from Baikonur, tend to drop their stages over Kazakhstan. Early British launches from Woomera in Australia let stage hardware fall in the Australian Outback (where some of it has been located and dragged to the tiny station of Williams Creek, where there is a very informal museum of crushed and destroyed rocket parts).
The problem with doing this, however, is that it means that for every launch, you're paying to build and fly and throw away a huge chunk of your rocket (in most cases, all of it except the payload). These things aren't cheap to make! They have very fine tolerances, require huge research and engineering efforts, and (mostly the engines) are very finely made, expensive machines. So, if there was a way to save and reuse these expensive bits, it would sure make space launch cheaper.
The United States Space Transportation System tried this. It didn't really succeed in making launches much cheaper- at least, not nearly as cheap as had been promised - because although it did reuse parts, it was also the most complex machine ever built by man, requiring astronomical (heh) amounts of maintenance for each use. Likely millions of person-hours of maintenance for each hour of actual powered flight, if you count everyone involved in the Shuttle program refurbishment chain. Despite the Solid Rocket Boosters and the Shuttle Orbiter itself both being recovered, it was incredibly expensive. And having to carry all the systems required to land up to seven people safely on a runway all the way up to orbit and back meant that it was spending most of its energy just flying (although the Shuttle did have a very good-sized payload - that's how the U.S. program mostly built the International Space Station, we had us a beast of a pickup truck).
Sorry, back to the ASDS. SpaceX has bet heavily that technology and material science has progressed to the point where it should be possible to not throw away the first stage (and, if they have their way, any stage) of a launch vehicle. After all, the booster for Falcon 9 has nine powerful engines on it, used to lift the entire fueled vehicle - but the specific impulse required to soft-land that one empty part is much, much less. So much less, that it can essentially reorient itself after separating from the second stage, perform a boostback burn on one of three of its nine engines, then perform a re-entry burn to bring itself back into the atmosphere, and, finally - a landing burn to soft-land on the surface.
Now, it's possible to bring a booster back and land it at the launch site. And in fact, SpaceX did that - their first successful booster recovery was made to a landing spot next to the launch site at Cape Canaveral Air Force Station - the booster essentially returned to its start point. However, in order to do so, the mission profile has to leave a lot more fuel in the booster at separation, because not only does it have to stop itself and land softly, it has to fly the entire distance it flew with all nine engines burning back to the beginning! And every drop of fuel and oxidizer used to make that happen is fuel and oxidizer that can't be used to lift the actual payload of the mission into orbit. So you are limited on what orbits you can reach, and how massive a payload you can put there.
If, however, the booster only has to brake to a relative halt (or even a slower return speed) and then land, you save precious delta-V. So for heavy payloads, or missions which require all the performance of the Falcon 9 system to reach their desired orbit, SpaceX have the booster instead re-enter the atmosphere and drop to the ocean, miles and miles offshore - much closer to where it was at MECO. Of course, that means you need something to land on! The Solid Rocket Boosters of the Space Shuttle would drop directly into the ocean - but they were very, very simple machines compared to the Falcon 9, with very few moving parts and no complex liquid-fueled engines. Even so, they required much effort to overhaul after their salt water dunkings.
The Falcon 9 is a much more complex and fragile machine. It can't go swimming. So SpaceX decided that if they were already designing the thing to land within a roughly two hundred foot square area, that area being on a ship wouldn't really bother the rocket much. But since things going wrong would inevitably lead to what SpaceX calls the 'Rapid Unplanned Disassembly' of the booster (read: big honkin' explosion as fuel and oxidizer gleefully self-immolate) well, then, you couldn't have people on this ship at the time.
Computer technology and Global Positioning System being what they are, it's now feasible to have a robotic ship whose only job is standing very very still until the booster lands on it, and then trying to stay as level as possible using thrusters and ballasting to make sure said booster doesn't fall over. This, then, is the ASDS, which is its job. It's an Autonomous (self-guided) Spaceport (okay, ambitious, but true) Drone (unmanned) Ship!
As icing on the cake for science fiction fans, Elon Musk and crew have named them after the Minds from Iain M. Banks' series The Culture. As a result, the first (experimental) ASDS was named the Just Read The Instructions. That ship, stationed in the Atlantic to recover launches from Canaveral, swiftly was superseded with a lessons-learned second version named the ASDS Of Course I Still Love You. After a long series of explosive failed attempts which caused wags to nickname the Falcon 9 "the most accurate anti-ship missile in history" it was this latter ship which, on April 8, 2016, first successfully received a landing Falcon 9 booster from the CRS-7 mission to resupply the ISS. The landing took place approximately 185 miles offshore, likely further from launch point as the flight path from Canaveral for the ISS intercept was to the northeast.
SpaceX has built a third ASDS, also named Just Read The Instructions. That ship has been sent to the Pacific coast to support missions launched from the West Coast of the US (Vandenberg AFB).
The point of all this, of course, is that a rocket which has been recovered may be useful for a second launch. If it is, then the cost of the booster - roughly 60% of the cost of the launch - has been cut in half, a savings of 30% of the total launch costs if you can believe SpaceX's numbers. If the second stage were also recoverable - perhaps after an orbit once around profile similar to the Shuttle's Abort Once Around - this savings could be even more dramatic. SpaceX is already building their Dragon spacecraft - the payload - with the intent of reusing them when possible.
The ASDS ships are McDonough Marine Service "Marmac" ocean barges (numbers 300, 303 and 304 at this time, I believe). They are modified by SpaceX and its contractors to include a 170-foot wide landing platform that extends for 245 ft of the barge's 300-ft total length. Although they are generally towed to their landing duty areas by a dedicated tugboat, the ASDS uses four 360-degree diesel engine thrusters for both propulsion, steering and station-keeping.
The ship can operate fully autonomously or via remote control from a nearby support vessel. Once a Falcon 9 lands on an ASDS, support crew board the ship and weld 'holding shoes' over the landing legs of the Falcon 9, implying that the deck surface itself is steel. Despite several highly dramatic direct crashes onto the decks of these ships, as well as landed boosters tipping over and exploding, the decks have
neverrarely shown any signs of any real damage. Edit As Kesper North has pointed out, there has been visual evidence of RUD damage to the ASDS Of Course I Still Love You at least once.
As for the future, the ASDS may not have shown us all its tricks yet. Back in 2014, SpaceX mumbled something about wanting to be able to refuel the Falcon 9 booster aboard the ASDS and have it *fly itself* back to Canaveral (!). I don't think they've mentioned this since, so it may have been hyperbole, or it may just be that they're going to keep schtum until they think it's feasible - after all, I'm not sure you'd want stores of rocket fuel and oxidizer about a ship which so far has had far more RUDs than landings. On top of that, time on the Falcon 9 engines is extremely expensive, and I can't imagine it'd be cheaper to do that then to just ferry the thing the 200+ miles back to the launch site. On the other hand, betting against SpaceX doesn't look like a good plan, so far. It's a pretty cool image, nonetheless!