What is the name of this logic circuit?
The overall diagram probably doesn't have a name. It shows two LEDs that are always going to light up together; if you were designing something systematically you probably wouldn't duplicated like that.  

(Of course duplication is sometimes what you want. A circuit to systematically duplicate a given signal probably does have a name.)

The individual components you're looking at are NAND gates. It's a neat fact that any logic gate can be built from NAND gates.
For example if you feed both inputs with a single signal (as happens several times in your diagram) then you've made a NOT gate.
Put a NOT gate on each of the inputs of a NAND and you've built an OR gate, which is what the first half of the diagram shows.

If we ignore the duplication that's happening at the end, then what we're looking at is a NOR gate made out of NAND gates. 

Such patterns might be useful if you want to keep a very narrow inventory of bread-board components, or you have a super-refined process for etching one single kind of gate into a silicon wafer.