widow line
A widow line is the closing line of a paragraph carried on its own to the top of a page, with everything above it left on the page before.
In more detail
Turn the page and the first thing on it is the tail of a sentence you have already begun — sometimes three words, sometimes one — with an expanse of white beside it. Then a blank, and only then does the next paragraph start. The top of the sheet, which is where the eye lands first, has been given to a scrap.
Its mirror image is the orphan line, a paragraph opening on a single line at the foot of a page instead of closing on one at the top. The two names get swapped from one workshop to the next, so what tells them apart is the position: the widow is alone at the top, the orphan is alone at the bottom.
Both are mended the same way, by moving the cut rather than the text. The break is walked back up the paragraph until a decent handful of lines is carried over, and if the paragraph is too short for that to work at all, the whole of it moves to the next page.
What it does on a real document
A program filling a frame stops when the frame is full and carries the remainder over, however small the remainder is. Left alone it will happily start a page with one word. The count is what stops it: how many lines of a paragraph have to travel together to the top of the following page.
It is asked for on the documents that run over several pages and get read by somebody who did not write them: conditions of sale, a lease, an insurance policy, a manual, a book. A sheet that opens on half a line tells that reader the document was assembled rather than laid out, and that is not the impression a signature page should give.
When the document asks for nothing, this engine carries 2 lines over. A document that wants three or four says so on the block of text concerned, and a table allowed to cut a row in the middle of its text says it on the cell whose text is being cut. The count is read inside one paragraph and nowhere else, so a list whose every line is a paragraph of its own is cut wherever the frame ends.
Where it comes from, and why
Printers have been chasing it since pages were made of metal, and the fault was noticed at the top of a sheet long before anybody thought to name it. The name that stuck came with the saying the trade used to keep the pair straight: a widow has no future, an orphan has no past.
What it costs is attention. A reader turning a page takes in the top of it before reading a word, and finding a fragment there means the sentence has to be reassembled from memory. Everything typography does about spacing exists to spare the reader that sort of small, repeated effort.