An accord is an agreement to accept a different performance in satisfaction of an existing obligation; the satisfaction is the performance of it. Together they discharge the original claim.
The disputed-claim requirement. Where the amount owed is genuinely disputed or unliquidated, acceptance of a lesser sum offered in full settlement discharges the debt. Where the debt is undisputed and liquidated, payment of less than the whole traditionally furnishes no consideration — though many states have modified this.
Full payment cheques. Under the UCC, a person against whom a claim is asserted may discharge it by tendering an instrument conspicuously marked as full satisfaction, if the claim is unliquidated or disputed in good faith and the claimant obtains payment. There are escape routes: a defined organisation can require such tenders to go to a designated office, and repayment within ninety days can undo the discharge.
Reserving rights does not help. Crossing out the legend and writing under protest generally does not preserve the claim in this context, contrary to widespread belief.
Practical handling. Any cheque arriving with settlement language should be routed to someone who will read it before it reaches the deposit run. Businesses that receive payments in volume should establish the designated-office procedure in advance, and state it on invoices.