They generally go as follows.

-Organisers post up the details on the schedule and rules, plus how to sign up, e.g. what information they require to contact you and to sort out assignments.

-You sign up, following their template for what info to give. This means committing yourself to write one story for another participant, usually with a word count minimum of 1000 words. Standard info that's asked would be what fandoms you're willing to write for (and for this, I guess what characters within those), what you're happy writing such as rating and genre, e.g. character study, angst, deathfic, as well not willing to writing, e.g. rating over PG-13, non-consensual, torture. Also important is what fandoms/characters/genres/ratings you want to receive, which are often combined with a prompt of some kind.

-Once signups are closed the organisers try to pair up participants based on the information collected, best matching what one person will write with what another wants, and then mail out assignments to people, giving out the recipients requests.

-You write to the recipients specifications, and word count minimum, though you don't need to meet all of them as long as you're avoiding their no-nos.

-Deadline comes around and writers should submit stories/story links to the organisers. Uusually a few people drop out or are delay, but how the organisers react to this varies. I've known some to delay posting of any until all are finished, though that tends to happen more with ficathons where they have anonymous posting of stories on the community/website and a later reveal of who wrote what.

The end result should be every participant ends up with a story meeting their specifications, though you can't account for stories being different than you imagine them but hopefully everyone is satisfied with their gift.

Sorry if that was too much detail...