Your teeth may not seem alive at first glance, but they have blood vessels and nerve endings which make them "alive".
While its hard outer coating, called enamel, may give off an impression that teeth are lifeless, they have living tissue inside. Blood vessels and nerves supply oxygen and nutrients directly into each tooth's structure as well as providing sensation when you bite into something crunchy like apple or an icy treat.
It is the outer layer that we typically think of as dead, and its cause can be complex. Enamel is made up of nonliving calcium phosphate crystals that cannot regrow like cells can. Rather, its formation during tooth development involves cells called ameloblasts which die.
Your enamel is a tough, calcified structure and beneath that lies dentin which contains living tissue with blood vessels and nerves running throughout it.
Dentin is produced by odontoblast cells originating in both the ectoderm and mesenchyme layers of tissue. These cells continue to exist within the pulp tissue throughout a tooth's lifespan, producing dentin tubules connected with its pulp tissue that are filled with water from within its pulp tissue. Each tubule can even sense hot and cold temperatures and adjust itself accordingly, helping regulate your temperature more effectively.
Cementum is a soft, bone-like material that covers the roots of teeth to help them stay in place and connect to fibers that support it, much like dentine but without blood supply or nerves. Cementum can move and reorganize with tooth movement so it is important that too much chewing or grinding doesn't damage this protective barrier.
The pulp of the tooth is the center of each tooth and contains nerves and blood vessels. This is a large reason of why teeth are considered "alive", because it contains blood to keep it healthy and can feel pain and hot/cold. When a tooth is damaged and the pulp dies off, this is what is considered as a "dead tooth". A tooth that is dying could require a root canal to prevent further infection throughout your mouth and even your blood stream.