How is local anesthesia administered?

Many many ways. Local can be given by nasal spray, skin cream, needle injection. Eyedrops, gargling, nerve block can numb all parts of the body including muscle, bone, tongue.
Via Injection. Local anesthetics are administered via injection, also called infiltration.
Injection. Local anesthetic is typically injected by needle into the tissue immediately below the skin in the area that one wants to numb. This can usually be performed with minimal pain for procedures that are performed at the level of the skin or just below. Muscle cannot be numbed by local anesthetic, limiting it's application for most operations.
No tube. Local anesthesia is given via a series of injections designed to deaden the area being worked on. Frequently a sedative is offered or given to make the experience more pleasant.