Note: For technical specs on this class of ship, see LordOmar's node USS Cole. The USS Cole (DDG-67) is, in fact, a DDG-51 class guided missile destroyer of the U.S. Navy.
LordOmar's node above has some good technical information on the DDG-51 class, so I won't recap that here. I will, however, take the chance to prattle on about the DDG-51 class, and the roles of ships in the U.S. Navy in general.
The DDG-51 is, according to the U.S. Navy, a destroyer. A destroyer is, typically, a ship which is designed for one particular task as part of a battle group. The term arises from the phrase/designation Torpedo Boat Destroyer, which is the original job of the class. They first appeared in the Atlantic during World War I, where they were tasked with protecting convoys and larger combatants from the predations of torpedo boats, some of which were also commonly known as submarines. They were also designed to take on the more classic type of torpedo boat; a thin-hulled, overengined, fast small craft designed to evade enemy naval gunfire and approach close enough to drop its load of two or more torpedoes before escaping. It was also small and cheap enough to be expendable, a fact not lost on their crews. The U.S. PT Boat (Patrol/Torpedo), the German WWII E-Boat, and the British S-boat were examples of the type.
In order to counter this threat, the Torpedo Boat Destroyer was born. It was a small (100-300ft) combatant, designed to be able to steam at high speeds and engage the maneuverable torpedo boats with small, rapid-firing cannon and machine guns. When submarines became a threat, it was a natural for emplacing depth charges and hedgehogs aboard to counter this new problem.
After World War II, the destroyer retained its primary task of ASW. As air-launched and sea-skimming missiles became common, however, modern naval task forces found that they suddenly needed a great deal more high-cap air defense, but had not much space/displacement available for the requisite weapons and ammunition storage aboard the larger ships, which were swiftly losing guns in favor of missile systems. Enter the destroyer. It had a plethora of small machine guns and autocannon which were perfect for close-range AA. As subhunting became a more complicated and difficult proposition with the invention of the nuclear-powered submarine, however, the destroyer finally ran into space and buoyancy limits - there simply wasn't enough of either aboard to do both. This was especially true as ships began to acquire helicopters; while hell on wheels against submarines, they weren't much good against air threats - and they took up an enormous amount of space aboard.
The U.S. Navy's response was to begin building specialized destroyers, and to relegate some of the sub-hunting duties to the frigate class. The FFG-7 and FFG-1052 class took up some of the mantle of convoy ASW, and destroyers such as the DD-963 Spruance class began to take over the blue water ASW and convoy ASuW duties of their predecessors the Farragut and Charles F. Adams class guided missile destroyers. The Spruances were backfitted with ABLs and VLS systems, allowing them to fire Tomahawk cruise missiles, as well as navy anti-air missiles such as the Standard. They were limited in their air defense capability, however, by their reliance on mechanical radars (in the older ships) and limited fire control resources in the newer Spruances.
The DDG-51 is the U.S. Navy's current answer to these shortfalls. It is a multi-mission ship, meaning it is designed to handle threats both on the surface and in the air. The installation of two MK. 41 VLS cells (fore and aft) means that the ship's missile load can be much larger than ships with box launchers or internal magazines and reloaders. In addition, the most significant weapon system aboard doesn't shoot anything but electromagnetic beams. The SPY-1D (spy-eye-dee, for the joke) is the radar system aboard which feeds the AEGIS fire control system.
The U.S. Navy perfected the AEGIS system aboard cruisers of the CG-47 Ticonderoga class. They managed, in a feat of shoehorning, to put it into the DDG-51 as well. The DDG-51 is, according to the navy, capable of AAW, ASW and ASuW - just like its larger cousins the cruisers. So how does it differ?
It displaces less. The CG-47 displaces approx. 9,600 tons; the DDG-51 around 8,300. That's not a huge difference. It's actually slightly larger - the CG-47 is 467 ft. in length, with a 55 ft. beam, whereas the DDG-51 is 466 feet in length with a width of 59 ft. The main difference (from the external view) is that the CG-47 uses a significant portion of its internal volume and displacement to house and support two ASW/ASuW helicopters, of the designation LAMPS (Sea Sprite), and in those after CG-48, the LAMPS-III, also known as SH-60 Sea Hawks. They also mount two Mk. 45 5"/54 gun turrets. The DDG-51 was designed around its VLS launchers, and has only one gun turret and no helicopters - although it does have facilities to service helos from shore or other ships while at sea (LAMPS-III electronics included), it doesn't have a hangar.
So what's the $*&*@ difference, really?
Not much, is the answer. The U.S. just likes calling them destroyers as opposed to cruisers, since we already had a cruiser class when they went to the drawing boards. And, if you count the lack of organic helo capability, they have slightly less ability to perform independent ASW/ASuW than the CG-47 - although the difference is marginal.
To put things in perspective, the JSDF - Japanese Self-Defense Forces - have their region's second most capable navy (second to the U.S.). Their navy, rather than having carriers, is built around their own AEGIS system ships - four cruisers which are slightly larger but really identical in characteristics to the DDG-51. They call them cruisers. The AEGIS system is bought/licensed from the US, and their hulls are native.
In any case, the DDG-51 is an enormously capable ship - the kind of ship that other nations really want in their arsenals if they have naval aspirations. They're cheap (in the U.S., compared to, say, Aircraft Carriers) and can do everything from hunt subs, to perform point or area air defense, to launch land strikes using Tomahawks. They are the 'all-purpose' combatant of the U.S. navy.