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.