Invoke the 25th Amendment! Wait, what IS the 25th Amendment?

There is increasing talk about the USA using the 25th Amendment to push out an increasingly unstable Donald Trump from the presidency. Talk about an impeachment has gone onto the back-burner recently for various reasons, but two Republicans who are looking to stand against Trump in the Republican primaries later this year (former Massachusetts governor Bill Weld and former Illinois Representative Joe Walsh) have openly mentioned the possibility.
But, for those of us who live outside of the US, what IS the 25th Amendment, and why is it being considered in the case of Trump, and what are the odds that it might ever actually be invoked?
The 25th Amendment to the US Constitution is a relatively addition, dating from 1965, in the aftermath of John F. Kennedy's assassination. It deals with the replacement of a president by the vice president in a case where the president is physically or mentally incapable of carrying out his duties as president. Generally this is done at the request of the president, as when Gerald Ford took over the presidency after Richard Nixon's resignation.
But Section 4 of the Amendment specifically allows for a president to be replaced by the vice president even against the president's wishes where the vice president and a majority of the sitting Cabinet secretaries (which, in the current 15 secretary setup, means 8 secretaries) deem the president to be "unable to discharge the powers and duties of his office". If the president disputes this (and, oh, just imagine Trump's reaction!) a two-thirds majority vote of Congress can maintain the decision, even against the President's wishes. Section 4 has never been enacted, ever.
Given the sycophantic current vice president, Mike Pence, and the current make-up of the Cabinet, such an invocation is very, very unlikely to come about (and would anyone really want Mike Pence as president anyway?) So, the recent calls for the 25th Amendment to be applied to Mr. Trump are not really serious policy initiatives, but electioneering sound-bites for his opponents, safe in the knowledge that they are very unlikely ever to have to follow through. They are just a cheap way of highlighting just how unstable and potentially  dangerous a second Trump term could be, for America and the world.