It is not a paste made from teeth.
Toothpaste is made from abrasives and detergents, usually with other minor ingredients to add special features. A simple toothpaste can be made from baking soda and salt. Both of these ingredients act as abrasives to help the toothbrush scrub off the film left by germs in the mouth.
About half of a typical toothpaste is abrasives. Typical abrasives used are sodium bicarbonate (baking soda), calcium carbonate (chalk), aluminum hydroxide, hydroxyapatite, and hydrated silica (opal). Hydroxyapatite is as hard as tooth enamel, and hydrated silica is harder than tooth enamel, so both of those can wear the teeth down more than the softer abrasives listed before them.
The detergents used in toothpaste are generally the same ones used in shampoo. They help clean your teeth in the same way as shampoo cleans your hair, and they create the foam when you brush.
You can’t taste the detergent because toothpastes are usually very strongly flavored, to overcome the taste of what the brush is removing from the teeth.
Tooth enamel can be strengthened against cavities by adding fluoride to toothpaste. Antibacterial agents are sometimes added to kill bacteria that form films of plaque. Sodium polyphosphate is added to toothpaste to reduce the formation of tartar (plaque that has become hardened by the calcium phosphate salts in saliva).
Whitening toothpastes contain peroxides that bleach the teeth.