In Civilization 5, one of the diplomatic options is a Pact of Secrecy. You sign it with one civilization against a specific other civilization.
There’s another called a Pact of Cooperation, which you sign with just the other civilization.
However, I can’t find any mention of either of these pacts in the manual or in the Civilpedia.
What do they do?
Edit: Added Pact of Cooperation, since its in the same boat and we have that answered too.
Both the Pact of Secrecy and Pact of Cooperation have been removed from newer versions of the game.
Since version 220.127.116.11 (15th December 2010) they have been approximately replaced with the Declaration of Friendship and Denounce, which have a similar meaning behind them, but are both now public and have more visible diplomatic repercussions.
Since version 18.104.22.168 (1st March 2011) both of these also expire after 50 turns.
Being a friend with another nation has the following effects:
Your relationship with the nation is improved (I think so, anyway).
The friend nation may occasionally request aid (eg, gold, luxury resource, etc) – if you refuse, there is a long term negative effect on your relationship.
If you befriend a nation, this will often have a positive effect on your relationships with other nations who are friends with the nation you befriend.
If you befriend a nation, this will often have a negative effect on your relationships with other nations who dislike the nation you befriend.
Groups of nations that are all friends with one another can effectively form a sort of unofficial alliance, and it can be powerful to be within a group of this nature. Similarly, be wary of attacking/upsetting a nation in a group, because you might find they drag all their allies in to the fight against you.