If my friend invited my dog andI into her house and my dog bit her dog, who pays the vet bill?

As the owner of the dog, you would be liable for the other dog's entire vet bill.  You could be sued for negligence.  Negligence is based on the failure to exercise due care (that degree of care that in this case a reasonable dog owner would have exercised under the same or similar circumstances) to prevent foreseeable harm.

In order for your neighbor to prove negligence, she would have to prove duty to control your dog, breach of duty, actual cause, proximate cause, and damages.

As a dog owner, you would have a duty to control your dog.  Breach of duty occurred when the neighbor's dog was bitten.  Actual cause means but for you not controlling your dog, would the neighbor's dog have been bitten?  If the answer is no, actual cause has been established.  Proximate cause means were there any unforeseeable intervening events that would relieve you of liability?  If the answer is no, proximate cause has been established.  Damages would be the amount of compensation the neighbor is seeking in a lawsuit (vet bill).  You may have a defense of assumption of the risk since the neighbor invited you and your dog into her house.  Assumption of the risk means that the neighbor recognized and understood the danger of allowing your dog into her house and voluntarily chose to encounter it. 

If there has been a previous incident in the past of your dog attacking, then you may be subject to strict liability.  Strict liability means you are liable whether or not you exercised due care.

If your neighbor files a lawsuit against you for the vet bill, it would most likely be a Small Claims Court case.  Also, if your neighbor fails to file a lawsuit prior to the expiration of the applicable statute of limitations, her case will be barred.