There are several different types of satellite orbits for space systems in orbit around the earth. The types vary in altitude, real or apparent path with respect to the earth, and duration of each orbit. For some missions, which use a constellation of multiple satellites, understanding the significance of the orbit means understanding the relationship of the orbits of the entire group.
Altitude further divides into the lowest (i.e., perigee) and highest (i.e., apogee) in the orbit. These vary the most with elliptical orbits and the least with circular orbits. A low perigee is valuable for missions in which sensors need to be close to the earth, or where low-power communications are being sent to the satellite. The disadvantage of the lower perigees is that they encounter more atmospheric friction, and, unless the satellite has fuel to adjust its orbit, gives it a shorter lifetime.
Angle of inclination is another parameter of the orbit. A 90 degree angle, which cannot be maintained, would have the satellite in a circular polar orbit over the North and South Poles. Zero degrees of inclination would be a perfect equatorial orbit.
Most orbits, helped by the rotation of the earth, travel east-to-west. It is possible to put a satellite in a west-to-east retrograde orbit, but the cost of the launch will be much higher, because it must cancel the earth's rotational speed.
Some satellites, such as those used for imagery intelligence (IMINT; taking detailed pictures) or signals intelligence (SIGINT; listening to things) carry substantial fuel, so their orbits can be shifted to areas of new interest. In some constellations, such as used for navigation or communications, there may be "in-orbit spares", so the spare needs to carry enough fuel so that it can shift into the orbit of a failed member of the constellation.
To think of these in fairly simple terms, assume they are circular and stay at a constant altitude.
Low Earth Orbit
Low earth orbit (LEO) can have a perigee as low as 100 miles, although 200-900 mile perigees are more common. The lower the perigee, the greater the atmospheric resistance, so, unless the satellite has significant station-keeping fuel, it will have the shortest lifetime in orbit, perhaps 5 years.
This orbit is best for listening to low-power transmissions aimed at the satellite or at a constellation, or for eavesdropping on low-power signals meant to stay on Earth. It also has the best view for taking detailed images of phenomena on Earth, for earth resources mapping or military reconnaissance. They are also important for communications where minimizing the round-trip speed-of-light delay to and from orbit is most important, as for satellite-based telephony.
Constellations need to have tens of satellites to provide continuous Earth coverage from this orbit. Individual orbits are of short duration, typically of 90 minutes.
Low Earth Orbit satellites must travel very quickly to resist the pull of gravity -- approximately 17,000 miles per hour. Because of this, they can orbit the planet in as little as 90 minutes.
Medium Earth Orbit
It should not be surprising to discover that medium earth orbit (MEO) is a compromise between the advantages of LEO and GEO (or even higher orbit). MEO orbits are longer in time than LEO, but, to an observer on the planet, still seem to move, and a constellation is needed for continuous coverage. The constellation will need fewer satellites than if they were in LEO, but many more than GEO.
At the lower altitudes, they can capture weaker signals than in GEO orbit, so this orbit may be more useful for SIGINT of stronger radar signals rather than weaker terrestrial communications. Their transmitter power and antenna size also represent a compromise between the modest requirements of LEO and the substantial requirements of GEO.
Geosynchronous Earth Orbit
Geosynchronous Earth Orbit (GEO) is an extremely valuable orbit, as it puts the satellite in an apparent "hovering" position over the equator, perfect for predictable communications relays. Getting a satellite into that orbit, at 22,300 miles/35,800 km, takes a large rocket and considerable onboard fuel. The general practice for reaching such orbit is indirect: the satellite initially goes into geosynchronous transfer orbit, from, at a precise time, it fires rockets to boost it to the higher altitude.
At this altitude, the orbit is 24 hours long, so the satellite appears to be in a fixed position in relation to the Earth. This is common for general-purpose communications satellites, although communications with low-power transmitters require lower orbits. In principle, three satellites, 120 degrees apart, can carry out a mission that requires continuous coverage of the Earth. Note that this makes orbital position, at least for satellites that would operate on the same radio frequency, a limited resource since only so many satellites can safely occupy the orbit without some danger of interference or even collision. Collision risk is greatest if there is an error in geosynchronous transfer.
In some cases, the satellites can transmit to one another directly, or via a specialized relay satellite.
Geostationary satellites, on the other hand, are in very high equatorial orbits above earth, which exactly matches the earth's rotation speed so that the satellite remains above a constant position on earth. To arrive at this orbit, the satellite is first launched to an orbit called geostationary transfer orbit, a highly elliptical orbit around the earth.
There are some useful if special-case orbits that change aspects of each orbit, such as apogee and perigee.
Highly Elliptical Orbit
Highly elliptical orbit (HEO) is especially useful for missions that require long coverage of the "high latitudes" near the North or South Poles. A common name for this orbit, Molniya, is the Russian for "lightning", and the Soviets/Russians gave high priority to such orbits to provide communications relay services in the high-latitude regions. Since much ground activity in Russia was in the Arctic, U.S. intelligence satellites would use HEO to give long coverage in those areas of interest.
Unfortunately, "highly elliptical orbit" conflicts with the abbreviation for high earth orbit. High earth orbit is a fairly ill-defined term, simply meaning part of the orbit is higher than GEO. To confuse things even more, part of Molniya orbits are usually above GEO, so Molniya (HEO) is a subset of high earth (HEO).
Both sides used the orbit for communications with military aircraft, such as strategic bombers, whose flight paths would go through polar regions. There are also quite peaceful uses, such as scientific analysis of polar phenomena, or communications for vessel monitoring systems on commercial fishing vessels in the high latitudes. Humans have cause to worry about military threats that would travel through these regions, while equally threatened fish remain blithely unaware.
The orbit varies from high to low altitude, and is sharply angled with respect to the equator, with 60 degrees being common. Typically, this results in an approximately 12 hour orbit, with a low, fast pass through the polar region that is not of interest, and a longer, higher, almost "hovering" position over the other pole. Still, constellations are required for continuous coverage.
This is another special case, usually in a low to medium orbit, where the satellite's orbit is symmetrical with respect to the Sun, not the Earth. Advantages of such an orbit include being able to pass a given spot on earth at the same time each day. For continuous coverage, constellations are still needed, but think of them as a bus lines where the buses arrive in a very regular schedule. Unless there are a large number of satellites in a earth-centric orbit, knowing which satellite will be overhead at a given time can take considerable computation, and having accurate parameters of its orbit -- which, in lower orbits, change over time.
Another advantage of being aligned with the Sun is being able to keep the satellite's solar panels in sunlight at all times. This is very useful for scientific payloads that are doing space research, and need continuous power.
Technical aspects of the orbit may require that only one part of the orbit, such as the perigee, seems constant with respect to the earth. Since the earth has movements other than perfect circular rotation, satellites of this type need station-keeping fuel to hold a constant orbit.