Planets are conquered by first bombing them to reduce the number of armies to a manageable level and then beaming down your own armies. A ship must be in orbit to bomb a planet. A side effect of this requirement is that shields are dropped and planetary defensive fire damages the hull directly. Planetary fire causes damage proportional to the number of defending armies. The actual formula is (armies/10) + 2 damage points twice per second.
Planets stats are updated every minute. There is a random chance that the number of armies will increase. There is also a random chance, that there will be a major die-off of armies (more likely on planets with many armies). If you have less than three armies on the planet, they will not grow as fast. Planets with no armies are owned by the independent team and will have no armies on them until someone beams one down.
Enemy planets do damage based on the number of armies they have. This means planets with no armies do no damage.
The teams' planets can provide fuel and repair services, which roughly double each operation's speed. Obviously, this makes these planets particularly valuable (and hence a lot of battles are fought around them). Fuel replenishes at an additional 2x rate when orbiting a planet which is marked as fuel sources. The same applies for planets marked as repair stations.
You can get information about any planet your team has orbited. However, if the planet is taken by any other team, you will lose information about the planet until it is reorbited.