The majority of LEGO parts can be classified into the following categories.
Bricks have a standard height of 1 brick and have studs on them.
Plates are 1/3 of a brick in height and have studs on them.
Tiles are 1/3 of a brick in height and have a smooth surface - i.e. no studs on top.
Beams are similar to bricks but have holes throughout them that fit Pins. Thin Beams are 1/3 the size of normal Beams.
Pins are used to connect with Beams and other connectors.
These are used with Pins and Axles to connect technic Beams.
Axles connect with some Technic Connectors
There are lots of further classifications for each of these parts, some of these are described below.
Similar to Beams, but with studs on top. Contains holes for pins/axles.
Bricks with a curved cloped.
Bricks with a printed pattern on them.
Bricks with a standard height but round.
Bricks with a smooth slope (various angles).
Bricks with additional features such as clips/pins/etc.
Bricks with slopes that come together in a wedge shape.
Plates with additional features such as clips/pins/etc.
Plates that are rounded or slightly curved with at least one stud.