(it's reddit so take it for what it's worth)
I'm a Foreign Service Officer currently working overseas. The ambassador in the country in which I'm stationed is still here and plans to come into work on Monday. He has not resigned or been dismissed.
FSOs are the U.S. diplomatic corps, and are career diplomats. There are also career diplomats with USAID, Treasury and other federal departments and agencies. There are a few ways to get in, but the way that most do (and what I did) was to sit for an exam, go to Washington for a day-long interview, pass medical and security clearances and wait. Mostly wait. FSOs spend their career on assignments in Washington and abroad that are usually two to four years long. About 70% of the U.S. ambassadors in the world are career diplomats who have spent many years as an FSO in Washington and abroad.
All U.S. ambassadors are nominated by the President, and then confirmed by the Senate. That includes the FSOs and the political appointees. Sometimes confirmation can take months, often not due to any fault of the appointee.
All ambassadors are required to offer their resignation to the President (or President-Elect) after Election Day. By custom, the President generally only accepts the resignation offers from the political appointee ambassadors and allows the career ambassadors to continue in their tour. In some presidential transitions, exceptions have been made on a case-by-case basis for certain ambassadors to stay after Inauguration Day, but President Trump and his team decided not to do that in this case, as is their right. In some cases, there are nominations for replacements, but in some cases, there are not at this point.
All embassies are led by the ambassador, and her or his deputy is called the Deputy Chief of Mission. DCMs handle the day-to-day running of the embassy. When the ambassador is out of country, the DCM acts as acting chief of mission. When there is no ambassador, as there will be in these countries in which political appointee ambassadors are gone, the DCM will be acting chief of mission for many weeks or months. During that time, the DCM is given the formal title of chargee d'affairs. Any DCM is going to be a well experienced FSO or other career diplomat, and many of them may be ambassadors at their next post.