if the Battery for a Laptop, has "Warranty: DOA" written in the sales specs, what does it mean?

I presume this is Dead On Arrival. If the battery is totally lifeless when you receive it you have a certain time period in which to claim a replacement. No point in calling them three months later to say it didn't work. They might still replace it, but you would be out return postage, packing charges, etc.

 

Please note this is a wild ass, shoot from the hip, guess.

Ophiolite is correct; the "Battery: DOA" portion of the contract means that if your battery arrives, or is replaced later and arrives and is not holding a charge, or overheating, that they will replace the battery for you. Usually the length of time will be specified by the computer vendor on the contract - the company I work for has a 1 year basic policy, or 90-day if it's a 3rd party manufactured, after that the warranty defaults to the Manufacturer. It kind of depends on who the vendor is.