A term in the contract that allows the buyer to inspect the property before the contract becomes binding. It is common for a buyer to seek concessions from a seller in exchange for satisfaction of the contingency, whether making certain repairs, or paying closing costs, or reducing the price, or whatever is acceptable to the parties.
The seller isn't obligated to anything. Any concession is at the seller's discretion. Although, needless to say, it is probably in the seller's interest to reach a compromise resolution and move on to closing. If the parties cannot reach an agreed resolution, the buyer can rescind the contract and get his earnest money back.