The classic slice
Thin, foldable, large triangular slices sold by the piece from a counter — this is the default "New York pizza" most visitors picture, and it's less about any one legendary shop than a genuinely consistent baseline quality across thousands of ordinary neighborhood spots. A good plain cheese slice, eaten standing at a counter, folded in half, is one of the most authentically New York things you can do for under five dollars.
Coal-oven and brick-oven pizza
An older, sit-down style cooked at very high heat in coal or wood-fired ovens, producing a charred, blistered crust. These are the places locals mean when they get serious about a "best pizza in the city" argument — sold as whole pies, not slices, and worth the wait at the well-known ones.
Sicilian and grandma-style
Thicker, square-cut styles — Sicilian is a taller, focaccia-like crust; grandma-style is thinner and rectangular, closer to a home-oven pizza than the classic coal-oven style. Both are widely available as an alternative to a classic slice, especially at neighborhood spots that serve both styles side by side.