Since the 1990s, the increasingly rapid discovery of extrasolar planets (planets orbiting other stars) has turned the traditional division of planets into rocky or terrestrial planets (like Earth and Mars) and Jovian planets or gas giants (like Jupiter and Saturn) from a handy shorthand for our own solar system into what seems to be an accurate classification of all known planets. Essentially, the difference between terrestrial and Jovian planets is that terrestrial planets are small and mostly made of rock, while Jovian planets are extremely large and mostly made up of gas.
A terrestrial planet is defined as one made up mostly of rock. In general, such planets consist of an iron core, sometimes molten, surrounded by a layer (or mantle, in geological terms) composed of silicates. They also possess atmospheres, but, unlike the gas giants, they have manufactured these geologically, through the gases spewed out by erupting volcanoes; and/or through the effects of massive impacts by asteroids and comets. In theory, such planets could also be carbon-based rather than silicon-based, depending on which elements dominated the solar nebula from which they emerged; however, no such planets have yet been found.
Terrestrial planets probably tend to be relatively small, because of the amount of mass they must contain as well as problems encountered with gravity by much larger objects; to the extent that they also orbit closer in to the Sun, these make them the most likely candidates in the search for extraterrestrial life. Whereas gas giants lack solid surfaces and probably possess gravity much too strong for recognizable life to emerge, terrestrial planets receiving the right amount of heat from their star could theoretically support life.
Within our solar system, the four inner planets - Mercury, Venus, Earth, and Mars - are all terrestrial planets. That all four should be in the same category indicates what variety can exist: Mercury really resembles our Moon more than Earth, while Venus is far hotter and Mars can get far colder than Earth. Some of the dwarf planets, like Ceres (in the Asteroid Belt), are also terrestrial, but the outer ones, like Pluto and Eris, and probably somewhat different, made up mostly of ice rather than rock.
Because all of our current methods for locating planets around other stars are biased towards heavy planets (simply because they're large, heavy, and therefore easy to find), we have had some difficulty finding terrestrial planets. Several roughly Earth-sized rocky planets have been found around pulsars, where they can be spotted when they occasionally block the pulsar's extremely strong, lighthouse-like radio signals. There is a growing list of so-called "super-Earths," whose mass is believed to be substantially higher than Earth's but substantially lower than our solar system's gas giants. Whether these are terrestrial or Jovian planets, we are not yet entirely sure. However, several, like Corot-7b and Gliese 581e, are only twice the size of Earth, and are almost certainly rocky in nature. NASA's Kepler Space Telescope is charged with finding Earth-sized planets around other stars.
JOVIAN PLANETS (GAS GIANTS)
Unlike the terrestrial planets, Jovian planets consist mostly (although probably not entirely) of gas rather than rock. They also tend to be extremely large; in the solar system, the lightest gas giant is 14 times the mass of the heaviest terrestrial planet, Earth. Virtually all of this mass consists of an enormous, multilayered atmosphere of hydrogen, helium, and, less commonly, ammonia, methane, and water. Although many popular sources claim they are entirely made up of gas, this is not entirely accurate: Jovian planets have a solid rocky core which is probably roughly the size of a terrestrial planet, surrounded by an enormous atmosphere of gas collected during planetary formation from the clouds of hydrogen and helium floating through the primordial solar nebula. There is probably no precise boundary between surface and atmosphere, but rather increasingly light layers of ices or liquids.
Gas giants are not believed to be good candidates to support life, because they probably lack well-defined surfaces with the necessary elements to support life. Moreover, because of their size, their extremely heavy gravity would be detrimental to life. (Although it is not inconceivable that some form of life could emerge which was specifically adapted to living in high-gravity environments.)
In our solar system, the four outermost true planets are Jovian planets: Jupiter, Saturn, Uranus, and Neptune. The first two of these consist almost entirely of hydrogen and helium, whereas the latter two contain larger proportions of the second set of gas giant chemicals (water, ammonia, and methane), which, at their great distances from the Sun, have probably frozen into ice.
Current techniques for finding planets around other stars require them to be large (so that their gravity causes a noticeable "wobble" in the star's own rotation, or a noticeable effect on starlight given a sufficiently powerful telescope). For this reason, the vast majority of planets discovered around other stars since the early 1990s are very massive, and thought to be gas giants.
Moreover, our techniques are also biased in favour of planets close to their stars (since these have even more pronounced effects), so the majority of such planets actually fit into a new category alien to our own solar system: the so-called "Hot Jupiters." Hot Jupiters are similar to our own gas giants, except that they orbit very close to their star, where the rocky planets can be found in our system. It's unlikely gas giants could form so close to their star; more likely, they would have migrated inwards from more distant orbits. Known "hot Jupiters" include Bellerophon (officially, 51 Pegasi b) and HAT-P-7b.