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A rocket is TSTO if it has Two Stages launching To Orbit from the earth's surface; as contrasting with SSTO, where all the hardware reaches orbit. There's even a concept referred to as One and a Half Stage to Orbit(!) which is more or less what The Space Shuttle uses- there, both stages fire at takeoff and then one of them drops away.

TSTO launches more payload, higher than SSTO can in most cases.

With TSTO the two stages separate slightly above the atmosphere, one part falls back to earth, and the other one lights a second engine and continues on up.

TSTO's can be either reusable (RLV), or expendable (ELV) or a mixture of the two.

The feeling in the industry is that TSTO might be a more cost effective approach than SSTO (see costs of launching to orbit).

Still, apart from costs, TSTO has some disadvantages. TSTO may be less safe, because there are more components to fail, they may be harder to test than SSTO which can incrementally build up to orbit, more the way aeroplane test programs work; and TSTOs require reassembly before launch, so they may be able to launch less often; and would require more support facilities.

Finally, they are a little less cool. Still, space launchers that are less cool, but a whole lot cheaper would be cool in a different way ;-).