It's slightly different depending on the party but essentially the Prime Minister can resign at any time.
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.
Yep. The PM is actually appointed by the Queen. But they have to demonstrate that they have enough support in parliament to form a working government. In practice this means that whatever party holds the majority gets to pick the PM. If there is no party with an overall majority (as in 2010) then two or more parties have to reach a coalition agreement.
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.