Skyrim Xbox 360 disc not working

You either have a disc with a manufacturing defect (which does happen on occasion), or the "brand new" game that you were sold was actually a pre-owned game. In either case, the solution is to return the disc for refund or replacement.

(The latter is highly likely since most physical-disc retailers really, really want to sell their stock of pre-owned games. This leads to some location managers to require their employees to use scummy tactics at the point of sale to quickly "inform" the customer that the game is pre-owned without actually making it obvious that they're not getting a brand-new disc.)