It depends on the circumstances. If you have an underage son, then there is a Regency until the son comes of age. If you have no son at all, the eldest brother becomes Daimyo. If no brothers, one of your Generals takes over. If no Generals, then the game generates a 'cousin' who appears and takes over. In all cases, having your Daimyo die in battle is not a good thing. I've heard that there can be some nasty penalties for a while if this happens, and apparently there are various penalties to order/diplomacy and such with all scenarios where you do not have a proper heir ready to take over, though I have never experienced them myself.