A type of computer strategy game (q.v.). The real-time strategy games involve control of military units in real time, often to perform military operations. They are called real-time games to separate them from turn based strategy games.
Many people consider Dune II to be the forefather of all modern real-time strategy games. After Dune II, there came to be precisely two game houses that got really known for the genre: Westwood (the original makers of Dune II), and Blizzard Entertainment. The two made many great games, and were very largely imitated elsewhere. (I think the only game to come close to popularity and quality of the games from these two was Age of Empires II and its followers, but the genre has a lot of sleeping gems buried under the mass of cheap clones...)
Westwood is these days best known for Command & Conquer and the related Red Alert series. Blizzard is known for the Warcraft series (Warcraft II is definitely the game that makes me remember them) and more recently the phenomenally successful Starcraft.
But all of the former games are nothing remarkable in terms of content. I realized this one day when I was playing WCII: It's nothing but a real-time version of Empire with pretty graphics. Everyone can beat the computer by building a lot of units and killing them - fortunately, in the games, the amount of available resources is limited.
The basic idea of "Dunesy" games is that they combine two vastly different games into one experience in a fundamentally incompatible way - and this is probably one of the reasons why turn-based strategy fans don't like them. The game includes a management / micromanagement side, where you capture and gather resources and secure them with your units, and you use the resources to build units. Then there's the tactical side, where you use your units to destroy opponent's units. Unfortunately for the Dune descendants the importance of micromanagement side is very, very high, and the games don't always depend the sense of tactics - they do depend, very heavily, on sense of micromanagement of your army and resource gathering. The tactics suffer in expense of the management, and so all of the actual tactical things are of lesser importance. People usually say that the only sin Blizzard ever commits is the "need to mess with little details on the battlefield" (The Finnish military jargon talks of "nysväys", I couldn't make a shorter translation in this context...)
Variants of the theme (hopefully better ones)
While the previously described kind of games have problems, the concept can be made to work! There are many games out there that already improve the genre a great deal, mostly by separating the micromanagement and economical things and the actual battle side.
A couple of good examples include Shogun: Total War and Medieval: Total War - these games have their own "turn-based strategy" side where you control the units on large scale and Build Stuff, and the real-time strategy side where zillions of sprite guys meet their bloody end on the fields of glory. Another example is the Myth series - pure tactics and bloodbath, you're given some units (possibly reinforcements later, too) and you're on your own. I have also heard people praising Total Annihilation for having a more working game model. There are also games like Battlezone (the new one, not the Atari arcade game) and the unreleased Golgotha that merge RTS components into first-person shooters.