The hinge clutch cover is the long black plastic part that covers the mechanical parts by the hinge of your MacBook, or the only plastic part visible to you at the fulcrum of your machine. It's visible from the back when your MacBook is closed and also when it is open at the base of your screen. Over time, the clutch cover can become cracked and eventually even break, exposing some part of the internals of your machine. This typically doesn't affect the operation or functionality of your computer. However, in addition to aesthetics it serves the purpose of keeping dust and other undesirables out of your MacBook.
The clutch cover on my 2011 15 Inch MacBook Pro eventually cracked. I purchased a new clutch cover from amazon for under $17.00 USD, and replaced it in less than an hour.