@CrossRoads is sort of correct, but is off on the details. The comments cover some of it, and I’d like to clarify and add some detail.
For purposes of airmen certification aircraft are divided into Categories and within Categories there are classes.
14 CFR §61.5 Certificates and ratings issued under this part.
(1) Aircraft category ratings—(i) Airplane.(ii) Rotorcraft.(iii)
Glider.(iv) Lighter-than-air.(v) Powered-lift.(vi) Powered
parachute.(vii) Weight-shift-control aircraft.
(2) Airplane class ratings—(i) Single-engine land.(ii) Multiengine
land.(iii) Single-engine sea.(iv) Multiengine sea.(3) Rotorcraft class
ratings—(i) Helicopter.(ii) Gyroplane.
A typical student pilot will train in an Airplane with a single engine that is equipped with tires—not floats. When pass their checkride they get their certificate with a Private Pilot, Single-Engine Land rating.
Once they have their certificate, they may begin to fly different airplanes that need endorsements in their logbooks. CFR §61.31 covers these. You would need an endorsement in your logbook to act as PIC for an airplane with over 200 horsepower; a complex airplane—controllable pitch, flaps, and retractable gear; or with a tailwheel.
A pilot may also get additional ratings to fly multi-engine aircraft, seaplanes, etc. Note that ratings require additional training and testing by a DPE while endorsements require training and logbook endorsements by a CFI.
A type rating is required by §61.31 for Large aircraft (I think this means over 12,500 lbs), turbine-powered aircraft, and other aircraft specified by the FAA.
Earlier I mentioned that when you pass your checkride you will have a Private Pilot rating. There are several other ratings that apply to your certificate, starting with your Student Pilot rating.
14 CFR §61.5 Certificates and ratings issued under this part.(a) The following certificates are issued under this part to an applicant who satisfactorily accomplishes the training and certification requirements for the certificate sought:
(1) Pilot certificates—
(i) Student pilot.
(ii) Sport pilot.
(iii) Recreational pilot.
(iv) Private pilot.
(v) Commercial pilot.
(vi) Airline transport pilot.
(2) Flight instructor certificates.
(3) Ground instructor certificates.
Sport pilot and Recreational pilot certificates have some limitations on the type of aircraft you can fly and require logbook endorsements for some privileges (e.g. flying in Class B for recreational pilot).
The instrument rating (§61.65 Instrument rating requirements.) is kind of odd in that it isn’t listed in the types of certificates, but it is an add-on to your pilot certificate. So a commercial pilot usually has an instrument rating and a CFI must have an instrument rating in order to qualify as a CFI.
Completely separate from all of this, aircraft themselves are classified as well.
Certification of Aircraft is covered by PART 23–AIRWORTHINESS STANDARDS: NORMAL, UTILITY, ACROBATIC, AND COMMUTER CATEGORY AIRPLANES Among other things it sets out the four types of aircraft that are normally flown in Part 91 operations.
14 CFR §23.3 Airplane categories.
(a) The normal category is limited to airplanes that have a seating configuration, excluding pilot seats, of nine or less, a maximum certificated takeoff weight of 12,500 pounds or less, and intended for nonacrobatic operation. Nonacrobatic operation includes:
(1) Any maneuver incident to normal flying;
(2) Stalls (except whip stalls); and
(3) Lazy eights, chandelles, and steep turns, in which the angle of bank is not more than 60 degrees.
(b) The utility category is limited to airplanes that have a seating configuration, excluding pilot seats, of nine or less, a maximum certificated takeoff weight of 12,500 pounds or less, and intended for limited acrobatic operation. Airplanes certificated in the utility category may be used in any of the operations covered under paragraph (a) of this section and in limited acrobatic operations. Limited acrobatic operation includes:
(1) Spins (if approved for the particular type of airplane); and
(2) Lazy eights, chandelles, and steep turns, or similar maneuvers, in which the angle of bank is more than 60 degrees but not more than 90 degrees.
(c) The acrobatic category is limited to airplanes that have a seating configuration, excluding pilot seats, of nine or less, a maximum certificated takeoff weight of 12,500 pounds or less, and intended for use without restrictions, other than those shown to be necessary as a result of required flight tests.
(d) The commuter category is limited to propeller-driven, multiengine airplanes that have a seating configuration, excluding pilot seats, of 19 or less, and a maximum certificated takeoff weight of 19,000 pounds or less. The commuter category operation is limited to any maneuver incident to normal flying, stalls (except whip stalls), and steep turns, in which the angle of bank is not more than 60 degrees.
(e) Except for commuter category, airplanes may be type certificated in more than one category if the requirements of each requested category are met.