I've spent a load of time looking for the answer to this question, but I cannot find it. What is the official name of this connector?
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It's a female crimp terminal (sometimes called an open barrel or u-barrel crimp connector). More specifically, since it has the plastic housing, it's called a PCB female crimp terminal.
Seems like that information was available from where you apparently got the image on ebay
If the case was that you happened across this image out of context, I recommend images.google.com. Google lets you search by image, basically you can bypass the 1000 words by just uploading the image.
The metallic parts are called crimp connectors - typically sold in strips such as in the photo, and removed from the strip before / after crimping to wire.
The black plastic parts are called single row crimp connector housings. They would be specified with pitch (distance between centers of each slot) at the least, with the common ones being 0.1" single row crimp connector housings.
The housings are available in singles, twos (as in the image in the question), threes, and so on, as well as in single, double, and even triple rows, maybe more.
At http://uk.rs-online.com, search for M20-1060200. They call it "M20 CRIMP HOUSING, SIL, 2w, 2.54mm"