Travertine is a kind of sedimentary stone that is common in Italy. Like marble, travertine is a form of limestone. When limestone is put under a lot of pressure for a long time (thousands of years), it can turn into travertine, the way peat turns into coal. If the travertine is left under pressure for thousands more years, it can turn into marble, the way carbon can turn into diamonds. So travertine is a kind of stone that is about halfway in between limestone and marble. Travertine is finer than limestone, but not as fine as marble.
The Romans used travertine on the main parts of buildings where it was too expensive to use marble. Often, for instance, the steps might be made of travertine while the columns or sculptures were made of marble. We do the same thing today.
Some examples of Roman buildings in travertine are the Colosseum in Rome or the Theater of Marcellus. Architects also used travertine in the Middle Ages, for example to build the Romanesque baptistry at Pisa, in Italy.
Learn by doing: find a building in your city that uses travertine
More about limestone
More about marble
Houses, Villas, and Palaces in the Roman World, by Alexander McKay (reprinted 1998). A standard text on Roman houses.