What type of router device sends client requests to the first available server in a server farm

A server farm is a group of computers connected together. There can by thousands of them in single farm. They are used to accomplish tasks or problems which can not be processed by a single personal computer.

Mid- tier server

Server Farm is the most common term, I think. Data Center is another term, but I think that may be a specific type of Server Farm.

Yes, but in this context, the word "farm" is used by one of its secondary definitions, which is "...an area of land devoted to the emplacement of a group of devices: e.g. a tank farm, an antenna farm, a server farm..." In this case, it's a "wind farm".