These are various ways of extending a dining table.
Butterfly: This is where the table can be pulled at one end to reveal a gap in the middle, into which an extra leaf, usually hidden beneath the table, can be raised, either manually or automatically so as to extend the surface area of the dining table
Drop leaf: This is a table which has a static central section, on either side (or just one side) of which there is a hinged section which can be raised to make a larger table. If these hinged sections are held in place with a bracket that feels like it clicks into place when you lift the hinged section up then this is a drop leaf table.
Gate leg: This is very similar to a drop leaf table, but rather than the hinged sections being held up by brackets, a leg swings out to support it