According to Islam, it is not ideal to have a pet dog, and it is better in terms of Islamic etiquette not to have a dog in the house, but it is not forbidden to have one.
However, if you have a dog in the house, you should be sure to observe Islamic rules of cleanliness for prayer. For instance, before praying salat, wash any area of yourself that the dog might have been licking or if the dog came into contact you while it was wet. When praying salat, avoid praying directly on the floor and instead pray on top of something (like a prayer rug or, if you do not have one, a towel or mat or something similar), and put it away when you are not using it so the dog cannot get to it. Also, be sure your clothes do not have dog hair on them and have not been licked by the dog or touched by a wet dog; if your dog is very affectionate, you may wish to have a separate pair of clothes that you only wear for salat (if this is possible). Also, if your family is very casual about dishes and where the dog goes, be sure the dishes you use have not been, ahem, slobbered on, licked, etc. (Some people are very strict about these things and some people are casual, that is why I am mentioning it)
Yes, this is inconvenient, which is one of the reasons it is better not to have a dog in the house! But you can still practice Islam and all aspects of Islam with or without a dog present.