I removed the seats from dining chairs I just bought but have never seen this type of construction. What is it called?
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I agree with the other answers that it's best called a finger joint or maybe a finger jointed corner brace. You could also call it a box joint if you want to distinguish it from the tapered finger joints that are sometimes used to join boards end-to-end in stock made up of shorter pieces.
The point of a corner brace is to add support to the corner joint. Usually, the brace is connected to each rail with glue and screws. Using a finger joint here adds a lot of glue area in a shearing direction, so the connection between brace and rails should be very strong even without mechanical fasteners. Forming that joint in a production setting would be fast and cheap, and it would save the cost of maybe 16 screws per chair and the time needed to drive them.
I found another example at dutchcrafters.com, which describes these braces as "grooved":
There is no official name for many more complex types of joints, and once you get to combining different types of joint techniques you can only really do your best to describe them succinctly.
There's a really great woodworking glossary website here: http://www.hobbithouseinc.com/personal/woodpics/_g_A.htm which describes lots of types of joint, but even he gets to the point where he's calling things something like an "angled keyed dovetail half lap joint". It is probably the most accurate way to describe the particular joint he's talking about (roughly half way down the page) but there's not a single person who, if asked to joint two pieces of timber using an angled keyed dovetail half lap joint, wouldn't ask you to explain with a drawing.
"Interlocking corner brace", "finger jointed corner brace", "box jointed angle brace" and "grooved corner brace" all seem like reasonable names for the joint to me.