I don't know the offical definition of the alpha dog but basically its something like the head dog of the house hold. My first yorkie has always had a very dominant personality and I thought she may end up being the alpha dog when i brought home the two puppies and she has definitely taken that role. She is the Queen of this yorkie house hold. Leader of the pack :-)
If you end up with two dogs that have very strong domaneering personalities (who each want to be THE alpha dog) than they may have issues getting along. Unfortunately my friend ran into this and ended up giving one of her dogs away after months of trying to get them to get along. They were actually being very mean to each other and purposely trying to cause each other harm.