My Basset Hound Has A Lump Under His Neck, What Is It?

This lump, like any new lump, needs to be examined by a veterinarian.  A fine needle aspirate should be done to determine if it is an abscess (infection), tumor (benign or malignant), lymph node, or salivary gland.  There are several lymph nodes in this area--they can become enlarged secondary to infection or inflammation of the face/oral cavity.  Also lymph nodes can be enlarged secondary to cancer--called lymphoma.  Lymphoma can affect dogs of all ages--it is not just a disease of old dogs.  This needs to be ruled out.  Based on the diagnosis your veterinarian can recommend appropriate treatment.

My basset hound puppy has had a small lump on the underneath of his neck since he was a small puppy, but as he has grown the lump has gotten larger. What could it be?

Sounds like a tumor to me is it hard or squishy type lump if its hard i would say a tumor if it's squishy could be a cist don't know if i spelled that right

If there is an infection, or was one, in dogs it will drain and create a pocket in their skin. It needs to be drained and probably needs antibiotics.