(Yes, I know I could Wiki it. For others browsing the thread, however, 2-5 sentences on how it all works would be useful.)
In terms of the Conservative Party they don't elect a deputy leader who would take over hence the internal party election for a new Prime Minister in October.
The Conservative Party also have processes to sacrifice their party leader (via the 1922 Committee) which allows the parliamentary MPs to call for a vote of no confidence in the leader and decide new candidates. This is how Margaret Thatcher was forced out.
So each party, as long as it holds the majority, can do whatever it likes to pick a PM?
He's resigning (or rather has announced he will resign before October) because he had certainly given the impression to his party that if 'Remain' didn't win, he'd resign. He could try and stay on, but the party might get rid of him and vote to make someone else leader (therefore forcing him to resign as Prime Minister).
Do PMs that leave go back to being house members? Seems like that would be kinda weird.
Yes.
> Does the house pick one of its own members for the job, or does the majority party get to pick anybody at all that they want to be PM?
Officially they can pick a member of either house of parliament. In practice picking a Lord would be extremely controversial and probably lead to some rapid constitutional reform.
> Do PMs that leave go back to being house members? Seems like that would be kinda weird.
They usually won't stand in the next election, and go off and do something else - either private business or charity, or one of the less direct parts of government (the house of lords, embassies or the like - traditionally Europe was another destination but I guess no longer). In theory they could stay on and work as foreign secretary or whatever for the person who replaced them (or return to the back benches) but yeah that would be weird.
Yep.
> Does the house pick one of its own members for the job, or does the majority party get to pick anybody at all that they want to be PM?
In most countries (the UK included) the majority party will pick one of their MPs. Australia once had a PM die in office (Harold Holt) and have a person in the Senate picked as the replacement, so the replacement ran in the byelection for the previous PM's seat.
> Do PMs that leave go back to being house members? Seems like that would be kinda weird.
Yeah, but they've also been house members the whole time they were PM, so it's not that wierd.
The plan is to sit around until the Tories find a new leader and try to hand over gracefully.
Whether there will be a new general election is also not part of the referendum, but it's a thinkable possibility.