The 1922 Committee is basically (although not exactly) the Tory Parliamentary Party, so if they led or supported a no-confidence motion the PM would be finished.
As it is, it's just one letter, and may end up being only a symbolic gesture. Still, I've felt for a while that May has only remained in Number 10 because kicking her out would be even worse -- well, worse for the Tories anyway.
Once 48 such ltters have been received a no confidence vote will take place among Tory MPs. This does not necessarily mean a leadership challenge should May win the vote. Even if she does win the deciding factor will be the number of those who vote against her as to whether she stays or goes. Anything over 100 would destroy what little credibility she has left.
From the Express 9th July:-
//Mr Brady is believed to be six letters short of the required number of letters to trigger a vote, according to the Sunday Mirror//
I grew up in the Shipley constituency. The M.P. used to be Sir Marcus Fox - chair of the 1922 committee and incredibly powerful - but approachable in some pubs! (Quite genial in fact.) As has been said, the number of letters received is vital, but it is rather more than that. If the support of the 1922 Committee is not forthcoming - the P.M. is dead in the water.
The 1922 committee represents backbench MPs and not government ministers. It takes its name from a committee of backbenchers formed in that year to withdraw the Tory Party from the Liberal - Tory coalition which had existed throughout WW I.