Assume you and a neighbor dispute how much money should cover accidental damage to his property (for example, you want to be neighborly and settle the matter of your goat getting loose and eating up his garden, or your kid broke a neighbor's window playing ball, or your husband ran over his mailbox, etc.).
Options may include doing nothing, involving authorities if available, and if also needed, suing in court. A better option may be to get a damage estimate or two (invite your neighbor to do the same) and offer to pay something in exchange for resolving the matter.
In the law sometimes this is referred to as an accord and satisfaction, a "method of discharging a claim whereby the parties agree to give and accept something in settlement of the claim and perform the agreement...." Black's Law Dictionary, 5th Ed.
It often involves tendering (delivering or mailing to the neighbor) a check for the amount you're willing to pay anyway, with a note in the "memo" section like "In full satisfaction of any claims regarding" the incident. The goal is to make clear (and in writing) the payment is being offered (and its acceptance/cashing constitutes agreement) to resolve the matter.
Your neighbor's cashing the check after writing on it "reserving rights" or some such generally will not avoid accord and satisfaction. As the Colorado supreme court has stated, "[i]n the case of a check offered as 'payment in full' for a disputed amount, generally a creditor [your neighbor] cannot avoid the consequences of accepting the accord, i.e., cashing the check, by declaring that he does not assent to the condition attached by the debtor [you]." Anderson v. Rosebrook, 737 P.2d 417, 419 (Colo. 1987). As renowned New York state Chief Judge Cardozo put it way back in 1932, "What is said is overridden by what is done...."
Practically, especially since checks generally are good for 6 months, it gives your neighbor something to think about for a while (a bird in hand is worth two in the bush), puts your money where your mouth is so to speak, may be the "neighborly" thing to do, and overall may be a good way to resolve things cheaply and quickly.
There are risks. Though rules of evidence may prevent it in court, "offering" to pay anything could be taken as a sign of culpability. Any amount offered may lead to more demanded. The law is uncertain as to whether this would be an enforceable accord and satisfaction in any one particular case (which tend to be fact specific). But, since the amounts involved usually are relatively minimal (these cases otherwise might end up in a small claims court), it may be an option worth pursuing.
As always, it is best to consult an experienced lawyer when considering options.