My therapy is psychodynamic psychotherapy and my therapist is an analyst.
I have been in CBT (and REBT) and found some useful things in them but it wasn't what I was looking for. I really like analysis. I like that each session you 'start where you are' and that anything and everything is valuable in your therapy.
My therapist is a MSW LMHC. I used to think "only" a psychologist would do. I found her by contacting a psychoanalytic institute in the nearest large city to me and asking for a referral to a candidate in my community. A candidate is a student who is already a therapist and is earning their psychoanalysis certificate. My therapist has been in practice for 30 years.
My therapy is modern analysis and is face to face rather me on the couch with her out of eyesight, although she is willing to do that if I that's what I want.

Here is a site that has a lot of information about types of therapies and types of therapists, and a good Question & Answer section: www.guidetopsychotherapy.com