Unlike some of the other modalities here, massage therapy is not intended to be a broad, overarching healing system. It’s an adjunct therapy that can be used alongside other therapies. (That is, your massage therapist is never going to be your primary care provider.) I think everyone knows what massage basically is, so I’ll focus on when and why it might be useful.
The most obvious benefit of massage is stress relief. This should not be underestimated, since chronic stress can wreak havoc on overall health. It may also be used to reduce muscle tension and/or pain. It seems to also have some farther-reaching effects, but these have not been clearly established through studies. The Mayo Clinic website has an overview of massage therapy that seems to be helpful and balanced. (In particular, the brief descriptions of the various types of massage are helpful.)
Myofascial release (“myo” = muscle; “fascia” = connective tissue) therapy is similar to massage. It consists of gentle, sustained pressure that essentially “stretches” certain portions of the muscles and surrounding connective tissue to work out knots and “kinks” and restore normal movement and function.
Visceral manipulation is a little like “chiropractic for your guts.” It involves the gentle manipulation of the organs, in much the same way as myofascial release gently manipulates the muscles. In this case, the organs and their associated connective tissue are being targeted.
Both of these, along with chiropractic therapy and other, similar manipulative techniques, are based on the principle that the function of the body flows from its structure, so when the structure is disrupted, the function is, as well. Restoring the structure is necessary to restore the function.