What is a Marriage and Family Therapist?

The short answer:  Someone who has education, training, and a license by the State to help people with their interpersonal relationships. These therapists may work with individuals, couples, families, and any combination. Here's the legal definition from the CA BBS.

There are different kinds of counselors out there to help. They differ according to education, training, and credentials. Here's some nomenclature to help you navigate the choices:

Certification and License
Certification means a practitioner took the time to learn something and demonstrated a certain degree of proficiency to get the stamp of approval from an organization that represents certain standards of the profession. Licensed means the person completed a degree program, participated in an internship to acquire supervised hours of experience, passed a state exam, and abides by ethical and legal guidelines of the profession. There is more professional accountability for someone who is licensed by the State. Insurance may also be more likely to cover services rendered by a licensed practitioner. In the end, someone's "credentials" may be less important than whether the experience is helpful to you or not.

Method
There are as many different methods of working with people as there are individuals. Interestingly, studies suggest that no method has been proven to be any better than another, but the studies do suggest that some therapy is better than no therapy!  [see article entitled "The War on Unhappiness," by Gary Greenberg, in Harper's Magazine, September 2010.]  Different ways of making contact with and working with your experience include talking, touch, movement, or artistic expression. Some practitioners help you look at the way you think about things; some practitioners work from a body-centered approach; some work with cultivating a sense of awareness; some use the relationship itself, between you and the therapist, as a tool for change.

Therapists, Psychotherapists, Marriage and Family Therapists (MFT, LMFT), Social Workers (LCSW), Mental Health Care Professionals, Counselors (MHC, LPC)
These could be different kinds of practitioners, or maybe not!  For example, not all Licensed Social Workers "solve problems through bureaucracy."  Many have private psychotherapy practices. So rather than elaborate about potential differences, I encourage you to contact a therapist directly, talk to him or her about what you are experiencing, ask how he or she might help, and see if you're a good match for each other. All these folks are just here to help you manage your experience and feel better.

Marriage Family Therapist, Marriage Counselor, Couples Counselor, Family Therapist
Like the group of terms above, these could be interchangeable. Or not! These folks are trained to help people with their interpersonal relationships. At best, they hold a perspective that includes both an overview of interpersonal dynamics as well as a light on each respective individual's inner experience. These folks are likely to work a little deeper than a Mediator or Pastoral Counselor or Coach, although some Coaches do help people experience change through profound inner connection.

Psychologists
These folks are distinguished from Master's level clinicians by having written a doctoral thesis as part of their education, based on some unique research they have undertaken (hence the Ph.D. after their name). They may also have been trained in administering certain kinds of psychological tests, although other counselors are permitted to administer tests as long as they are trained and competent to do so.

Psychiatrists
Unlike the practitioners above, these folks are licensed to prescribe drugs, just as Medical Doctors are. It has been my experience that Psychiatrists have a real knack for assessment, but LMFTs are also trained to "diagnose" and "treat" individuals with "mental illness." While making prescription recommendations is outside the scope of practice for Counselors and Psychologists, they are typically trained to know if and when it might be appropriate to discuss the option of drug use, and can help make an appropriate referral to a Psychiatrist. Therapists often work in conjunction with Psychiatrists, and the combination of therapy and medication management is usually more effective than just taking drugs.

Psychoanalysts
Think Freud and Jung. Think dreams, archetypes, and material making itself known from the unconscious to the conscious to be understood and healed. It is generally a longer-term process.