If the House of Commons votes No Confidence in the government then a general election follows (IIRC the last time this happened was 1979, resulting in the election of Mrs Thatcher).
But Jeremy Corbyn has said that he will propose a motion of No Confidence in the Prime Minister. Suppose this motion succeeds. What does this mean? Must the PM resign?
Jeremy Corbyn has proposed that there is a non-binding vote in Parliament on the Prime Minister. This is not the no-confidence motion that would lead to an election. The vote would have no required effect.
Corbyn is suggesting this for a couple of reasons. First, as Laura Kunnsberg suggested on the BBC, he does not believe that he could pass a no-confidence vote. As if the DUP merely abstain, the government will win. Winning a vote of confidence could bolster the PMs position. Secondly, he knows that there are a minority of the conservative parliamentary party who voted against May in a secret ballot. He wants to call them hypocrites if they don't vote against May in an open division in the Commons.
But such a motion does not have a precedent. In principle the Commons can vote on any motion it wants, but I would not bet on this idea ever reaching the floor of the house.
The Prime Minister of the United Kingdom is appointed by the Queen, as "the person most likely to command the confidence of the House of Commons". (Notably, it is not an Elected position, but being elected to other positions - such as Leader of your Party - are likely to make you appear more eligible to Her Majesty.)
If the House of Commons voted "No Confidence" in the PM, then the Queen would most likely appoint a new Prime Minister, unless she had reason to suspect that all other options would command even less Confidence from the House of Commons.
While most Prime Ministers who received a vote of No Confidence of over 50% of the House would probably choose to resign, it would not be mandatory - but, doing so would be better than being dismissed by the Queen.
(Likewise, the Queen is perfectly free to say to a PM resigning in the result of, say, a 52% No Confidence vote "Well, you may only command 48% Confidence - but the next best candidate only commands 40%. I would like you to stay in the post.")
To clarify in response to comments: A Vote of No Confidence in the Prime Minister has no real impact. There is no precedent and no legislation attached to the situation, only a handful of conventions with links best described in various shades of "tenuous" - so, even if such a vote was passed, Parliament could theoretically choose to just shrug and ignore the result. Not necessarily a good or wise choice, but one that is available without breaking, bending or modifying any existing strictures.
As such, Mr Corbyn's proposed motion is pretty much just smoke and mirrors: it may sound impressive, but is unlikely to put him at risk of any blame if everything goes wrong - and I very much doubt he really expected it to be allocated any time for debate in the House of Commons in the first place. But (for many of the same reasons), as a Political tactic to get him seen as "doing something", and to achieve some resonance with dissatisfied voters, it's a fairly shrewd move.
From everything I've read, and the two existing answers, there appears to be no actual repercussions to the PM or the Government in losing this vote, if indeed it ever happens.
What it does allow is for the Labour party to take control of the narrative. Losing a vote of no confidence in the Government means that the chances of a General Election are significantly reduced and the Labour Party is currently strongly committed to getting that General Election. Trying and failing will be a disaster, and in the aftermath of such a vote all news cycles will focus solely on the failure to remove the government.
Losing a non-binding vote of confidence in the Prime Minister does not have the same effect. It allows the opposition parties to turn the spotlight on the 117 Conservative MPs who just last week demonstrated their lack of confidence in Theresa May. They believed she was not capable of running their party, but if they back her against Labour's motion they are saying the believe she is capable of running the country.
The Labour Party leadership may believe they can capitalise on this discrepancy to make the case that Conservative MPs are more interested what's good for them and/or their party than what is good for the UK.
