The short answer

Start by separating pattern from breed: tabby, calico, tortoiseshell and tuxedo are coat patterns found in many breeds and in cats with no breed at all. Then look at coat length, body build, head shape, ears and tail. If the cat came without papers it is almost certainly a domestic shorthair or longhair, and the useful question becomes which breeds it resembles.

Step 1: name the pattern, which is not the breed

What most people call their cat's breed is the coat. Get this part right and half the confusion goes away.

Tabby cat: Tabby is a coat pattern, not a breed.
Tuxedo cat: A tuxedo cat is a black and white bicolor, and bicolor is a pattern rather than a breed.
Calico cat: Calico is a coat pattern, not a breed.
Tortoiseshell cat: A tortoiseshell is a cat whose coat mixes red with black, chocolate or seal in a mottled pattern with no required white.
Lynx point Siamese: A lynx point is a pointed cat whose points carry tabby markings instead of solid color.
Orange tabby cat: Every orange cat is a tabby.
Torbie cat: A torbie is a tortoiseshell with tabby markings inside the color.
Dilute calico and dilute tortie: Dilute versions swap the colors, not the pattern.

Step 2: coat, build, face, tail

Coat length and texture. Short, semi-long or long; single or with a dense undercoat; any curl or wave.
Build. Long and fine-boned, stocky and round, or somewhere in the middle, where most cats sit.
Head and ears. Wedge, round or square muzzle; ear size and set; tufts at the tips.
Eyes. Shape and color. Blue eyes with a pointed coat mean something; green eyes on a gray cat mean less than people hope.
Tail. Full plume, whip, bob, or kinked.

Step 3: check the breed you are hoping for

Big and fluffy does not make a Maine Coon, and gray does not make a Russian Blue. These pairs get mixed up most.

Is my cat a Maine Coon?: Probably not, and that is the ordinary answer rather than a disappointing one.
Domestic shorthair: Domestic shorthair is not a breed.
Calico vs tortoiseshell: White is the difference.
Maine Coon vs Norwegian Forest Cat: Read the head, not the coat.
Bombay cat vs black cat: A Bombay is a registered breed with a written color requirement; a black cat is any cat with a black coat.
Russian Blue vs gray cat: Gray is a color that any cat can wear; Russian Blue is a registered breed with a written description.
Ragdoll vs Birman: Check the feet.

Photo, papers or DNA: what each can tell you

A photo shows resemblance. It is quick, and it is the right tool for the everyday question of what your cat is like. Papers are the only thing that make a cat a pedigree cat: a registry record of its parents. A DNA test compares your cat's markers against reference populations and can report ancestry signals and some trait and health variants. None of the three changes the cat.

Questions people ask

What is a domestic shorthair?

The name for a short-coated cat of no particular breed, which describes most pet cats. It is not a lesser answer. Domestic cats come in every pattern and color, and their mixed ancestry is why no two look quite alike.

Can a photo prove my cat is a purebred?

No. A photo can show that a cat resembles a breed. Pedigree is a matter of record: registration papers from a cat registry that trace the parents. A DNA test can add information about ancestry and some traits, and even that does not make a cat registrable.