You can put them on any planet after you research them. Each one gives a bonus to the credit income of that planet. Sadly, you can put as many as will fit around a planet (my last MP opponent put up to 5 around most of his planets, now THAT'S an exciting space battle tactic

). The longest "string" of planets that have at least one Trade Port will get a bonus (hover your mouse over the Credits counter at the top of the screen, and it will show a white line of the most profitable routes), and will also show a list of what Trade Ports are giving you credit-wise. Gravity wells that can't be colinzied don't count in figureing the best trade route.
There is no deminishing return on Trade Ports, so, again sadly, spamming as many as you can around planets is a good tactic. But, they do cost quite an investment at first. And you could be attacked with few defences if you spend all that money on Trade Ports (early game).