If I live in a condo with my family and the other night the washing machine broke and flooded our floor and the person below us, what is our liability?

Yes, your family is most likely liable, because it was your washing machine and it is most likely the case that the leak was due to excessive age (should have been replaced), overuse or misuse (such as too large a load), ignoring signs that the washer was breaking down (not running as well; small leaks), etc. Washing machines do not spontaneously break more often than not; usually, there was some warning sign or something done wrong. However, if you sued--and you don't have to pay anything unless and until you are sued and the other side wins (though sometimes it's a good idea to try to voluntarily settle for something you are willing and able to pay, to avoid the cost of defending a lawsuit and also the possibility you'd end up losing badly and paying more than you can afford).

If you think that the washer was defective, if it's not too old a machine (there's a statute of limlimitations, or maxiumum time period, for suing for product defects) and you are sued, you can bring the manufacturer in as a "third-party defendent"--someone you are suing in the same lawsuit in which you are being sued--and try to either show that it was their fault, not yours, so that they pay the other party directly, or try to recover from them some portion of anything you are ordered to pay. You can also try to simply defend the lawsuit if sued, by trying to show that your family did nothing wrong (i.e. the machine wasn't so old that it should have been replaced; there were no warning signs you ignored; it wasn't misused) and that you are therefore not liable, or financially responsible.

The warranty will not help with a lawsuit: it might get the machine itself repaired or replaced, but has nothing to do with your liability.