For full details of the process, see the announcement on Meta Stack Exchange. The timeline:
On April 19, if there are three or more candidates, we'll run an election. If not, I'll simply appoint the candidates. (There's a small chance we'll need to remove a nomination, but I doubt that will come up.)
If there is an election, I'll announce the results on meta on April 27.
(Note for current moderators: there's no need to nominate yourself even though you'll likely get an email saying you should. The system assumes the first election is a graduation election, which would mean moderators would need to be re-elected. This isn't that sort of election.)
If you have any questions about the process, please stick them in an answer here.