First, lets talk about simple (near to) circular orbits. Usually, you can divide them into Low, Medium and High.
All geocentric orbits with an altitude up to 2,000km (1,240mi) are classified as a Low Earth Orbit (LEO). Any object that is supposed to stay in space for a longer period of time (such as satellites or the ISS) usually has an altitude of >300km, since orbits below that altitude would be impractical due to atmospheric drag in the Thermosphere, causing orbital decay.
Although the altitude of the ISS is higher than that, it still loses ~50-100m a day and has to do orbital boost burns every so often.
LEOs are better than higher-altitude orbits because they require less Delta-V to reach, so it is cheaper to build space stations there. They are worse because you can see objects in LEO only from a small portion of the Earth's surface.
(Geocentric) orbits above 2,000km to just below geosynchronous orbits (35,786km) are classified as Medium Earth Orbits (MEO), "most commonly at 20,200 km or 20,650, with an orbital period of 12 hours", as used by the GPS satellites.
MEOs are good because with an orbital period of 12 hours, you can easily calculate when the satellite is going to pass over you and you still have a reasonable viewing angle.
(Geocentric) orbits above 35,786km are classified as High Earth Orbits (not HEO!). Weather and communications satellites tend to be in High Earth Orbits because they can "see" a large portion of the earths surface from this altitude.
Some examples for interesting elliptic orbits are the Tundra and the Molniya orbit.
Geosynchronous orbits obviously only work when placed above the equator. If you want to have a satellite stationary over very high latitudes, they don't work.
So instead you use one of the two orbits. They are highly elliptical and have an inclination of ~63.4°. The high eccentricity means that a satellite has a long apogee dwell, so it remains around a certain point on the surface for most of its orbit, then quickly swoops around the earth and comes into vision again after a short time.
The Tundra orbit has a perigee of 1,000km and an apogee of 70,580, giving it an orbital period of 24 hours. The only current user of Tundra orbits is the Sirius Satellite Radio. It uses a constellation of three satellites, with which it can provide all of North America with its stream for 24 hours a day.
The Molniya orbit has a perigee of 1,000km and an apogee of ~40,000, giving it an orbital period of 12 hours. They were primarily used by the Molniya communications satellites and some spy satellites during the Cold War.
Another point to consider is that orbits intended for manned spacecraft shouldn't cross the Van Allen radiation belt (although that will be hardly avoidable when going to the moon), so that excludes orbits of around 1,000-60,000km from the list for manned craft.