The ratio of troops you need to the population is completely dependent on the tech level of your troops. The more powerful they are, the fewer you need. It doesn't hurt to have too many, though.
The population of a planet will produce one "militia" per twenty million inhabitants. This ratio is set in SETIINGS.TXT though, and you could change it if you wanted to. Also note, the bigger the capacity of the transport you send in, the better. At least, with fully populated worlds. You are thrown into ground combat immediately after the troops are dropped (still) and if you don't drop enough to win they are all killed off, although you can make ground combat more "drawn out" by changing the number of rounds in SETTINGS.TXT so it has more of a chance to run over into another strategic turn. This gives you some chances to drop more troops.
Also, the "odds" seem to have a real effect on the over-all combat. If your troops are badly out-numbered they lose quicker, and if they out-number the defenders by some large ratio they win quicker with fewer losses. Finally, the presence of troops on a planet affects "happiness" levels now. So you want to leave an occupying force to make the population realize they now have to be happy with your rule.
Note: You don't "get the ability to breathe their atmosphere" you get the POPULATION which has the ability to breathe that atmosphere. I.E. - slaves! There are other ways to do this. Research 'Ship Capture' and take some colony ships (or even population transports if you can catch one). A capture ship with stealth armor or even a cloaking device is really cool to have.
[This message has been edited by Baron Munchausen (edited 08 January 2001).]