Excellent question! The planets in our solar system are each made out of different things. The inner planets (Mercury, Venus, Earth, and Mars) are relatively small rocky planets made up mostly of silicate minerals and iron and nickel metal. Most of the earth is made up of only three elements: iron, silicon, and oxygen. Most of the iron is in the core of the earth. Most of the oxygen is locked up in minerals with silicon. In fact, the mineral quartz is made up of only silicon and oxygen. The inner planets are relatively small and mostly solid, though earth has a thin atmosphere made up mostly of nitrogen and oxygen gases.

The outer planets (Jupiter, Saturn, Uranus, and Neptune) have much different compositions. They all have solid cores of rock, metal, and maybe even ice, but these planets are totally enveloped in thick atmospheres of gas. Hydrogen and Helium (the two lightest elements) make up most of the gases in the atmospheres of these planets. It is thought that light gases, such as hydrogen and helium, were “pushed” to the outer solar system by radiation and matter streaming from the sun (Grotzinger et al., 2007). Smaller, solid moons made out of rock and ice orbit the large gaseous outer planets.

references
Grotzinger, J., Jordan, T.H., Press, F., Siever, R. (2007). Understanding Earth. New York: Freeman and Company.

That depends on the planet. There are four kinds of objects just in our own solar system that are (or have been called) planets. These are:

1. Overgrown asteroids (Mercury, Venus, Earth, Mars), which are made out of rock but have metal (probably iron) cores, and may have relatively thin atmospheres;
2. Giant comets (Pluto, Eris), which are various kinds of ice, e.g. water, methane, ammonia, carbon dioxide, and other things that freeze because it's so cold in the outer solar system (note: right now these aren't called planets, but I'm including them because they have been called planets in the past and may be again in the future);
3. Failed stars (Jupiter, Saturn), which are mainly hydrogen and helium compressed into liquid metallic form by the immense pressure inside of these planets, possibly with rocky cores as well;

4. In-between objects (Uranus, Neptune), which have thick clouds of water, methane, ammonia, etc. ices that turn into hot liquid (again due to pressure), but with large rocky cores in the interior.

In addition, other solar systems are thought to have:
5. "Ocean" planets, which are mainly composed of water but with rocky cores where the point where the air (itself water vapor) meets the sea is difficult to define due to pressure (incidentally, Ganymede and Europa might qualify as this kind of planet if there weren't moons orbiting Jupiter);
6. Only partially-failed stars (called brown dwarfs), which are big enough to burn deuterium (a kind of hydrogen), but not regular hydrogen, preventing them from being true stars;
7. "Pulsar planets", created from debris falling back on themselves after a supernova;
8. Rock-ice mixtures, containing roughly equal parts asteroid-like rock and comet-like ice, like some of the moons of Uranus;
9. "Proto-planets", dense clouds of dust and asteroids that will become planets but haven't collapsed completely yet;
10. Other things that we haven't even seen any evidence yet.
Stars can also orbit other stars - this happens a lot, in fact.