What are the requirements to have the option to sign a Pact of Research with another Civ?
Both sides need enough money for the agreement, and at least one side has researched Philosophy.
The cost for a research agreement scales with the era of the more advanced player and with the game speed:
You can also have only one active research agreement per civilization, and research agreements are not possible during war (as bwarner pointed out).
If the enemy civilization has not enough money for the research agreement you can also gift (or loan it against gold per turn) the necessary gold to them.
To enter a research agreement, both parties need to have the required amount of gold and both need to have researched Philosophy.