Quite often, graduate level research is done by researchers from different institutions, whether different universities or cooperation between a university and a private corporation.
How do the researchers get to know each other? How does the collaboration begin?
Go to a conference/workshop. Ask intelligent questions in talks. Talk to speakers afterwards. (Have your advisor introduce you if you're shy.) Eat lunch with them and their students. Hang out in the hallways. Talk about your research. Listen to other people talk about their research. Listen to gossip. Throw in a few dirty jokes. Grab some coffee, or beer. Scribble on napkins. Go out for dinner with the students. Drink. Play pool. Try that new absinthe place around the corner. Scribble on napkins. Throw up in the alley. Stumble home. Sleep. Grab coffee. Run into someone you met at the absinthe place. Commiserate about your hangovers. Talk about your research. Scribble on napkins. Have a good idea. Skip the next four hours of talks. Fly home. Exchange drafts over email/github. Chat on Skype. Revise, revise, revise. Submit to the next conference. Repeat.
Very important: Do not skip the first 15 steps.
The key is really to widen your connections and make yourself known, and this can be done in several ways. To socialize in meetings (a la JeffE) is certainly one way. Personally I have never found collaborators that quickly, sometimes you strike up a conversation (at a meeting or through other means) and maybe years down the line, the right idea comes up where this collaboration gets started. So as with many scientific discoveries, collaborations may start where you least exepct it and with persons you have known for long for other reasons. Hence making yourself known for good science may attract other persons to contact you as will you with those you can see common ground. Becoming a "known player" (however you want to interpret it) in the field is also likely to provide results in the long term. I would also add that you should listen and learn from colleagues about other people. I do not mean gossip, but rather the way other scientists are thought of. This will prepare you for contacting them. Another example of get known and know your surroundings. Finally, also remmeber that we are all different in how we interact and there is therefore no single way to strike up collaborations.
