The presidential nominee "chooses a running mate," and then the national convention delegates rubber-stamp that man or woman. There have been a few attempts to force a fight (my recollection is that Howard Dean supporters tried to get enough delegates lined up to force a competitive vote and make him the veep nominee whether Kerry wanted him or not in 2004), but for the most part it is a process of coronation by acclamation based on the presidential candidate's desires.
The Libertarian Party does things differently.
In any given election cycle, one or more candidates formally declare their candidacy for the vice-presidency long before we even know who the presidential nominee will be. They campaign. Maybe not as vigorously as the presidential contenders, but they do hold themselves out as candidates specifically for the vice-presidential nomination, and usually without trying to attach themselves to any particular presidential candidate.
The presidential nominee is given five minutes of speaking time to endorse a VP candidate or to share his thoughts on what he hopes the delegates will do, but it's not a coronation.
For example, in 2008, after the presidential nomination had been decided (the nominee was Bob Barr), two presidential candidates switched tracks and ran for VP. Barr endorsed one of them (Wayne Allyn Root), but it took two ballots for him to prevail over the other (Steve Kubby). Unfortunately, the previously announced VP candidates didn't do well in that contest.
This tradition of the LP's has been atrophying over the last couple of cycles.
First, as I mentioned, in 2008 a couple of presidential contenders jumped in at the last minute, pretty much undoing the work of those who had been going for the VP slot from the first. Nothing we can, or really should, do about that, I guess, but it's still a little sad.
Secondly, in 2012 two people -- Gary Johnson and Jim Gray -- ran as a kind of package deal and were nominated with neither muss nor fuss.
It would be good for the LP if we went back to REAL vice-presidential contests.
Right now, to the best of my knowledge, there's only one declared candidate for the Libertarian Party's 2016 vice- presidential nomination [thanks to Thane Eichenauer for the typo catch]. His name is Mark G. Elworth and you can read about him at Independent Political Report.
No, I'm neither endorsing Mr. Elworth nor urging you to vote against him. If you're this far into this blog post, you know how to read and should be able to make up your own mind as to whether or not he's what you're looking for in a vice-presidential nominee.
But I do think the party deserves more than one candidate to choose from.
If you've ever said to yourself "I'd make a good candidate for vice-president" -- or if you're saying that to yourself right now -- why not throw your hat in the ring?
Update: Thanks to Andy Bakker for letting me know that Kerry Douglas McKennon is also a declared candidate for the VP nomination!