The concept is quite simple. A larger, wealthier nation (no offense meant) sends you the set price (1 mil, 3 mil, 6, etc.) and in return you send them however much tech they request. It'll take 10 days until you can send the tech to them, then another 10 days until you can use that foreign aid slot again.
I'm not sure what the current rates are, to be honest. It was (almost) always 3 million for 50 tech, but I think it may be 3 million for 100 tech now. I think the only requirements for you to receive aid are that you have to have war as an option. I'm not sure if government type affects it or not.
If you want a good, solid growth program (tech deals and trade circle galore), find a strong alliance that offers those things. You ought to have received 10 million recruitment messages by now, I'm positive you've received at least one message that offers that.
Hope that helps!
-Alasdair