Definition

A quadrilateral is a polygon with exactly four sides.

In the words of Euclid:

Rectilineal figures are those which are contained by straight lines, trilateral figures being those contained by three, quadrilateral those contained by four, and multi-lateral those contained by more than four straight lines.

Because it is a polygon, it follows that it also has four vertices.

Square

A square is a regular quadrilateral.

That is, a regular polygon with $4$ sides.

That is, a square is a plane figure with four sides all the same length and whose angles are all equal.

Oblong

An oblong is a quadrilateral whose angles are all right angles, but whose sides are not all the same length:

Rectangle

A rectangle is a quadrilateral all of whose angles are equal to a right angle, and whose sides may or may not all be the same length.

Parallelogram

A parallelogram is a quadrilateral whose opposite sides are parallel to each other, and whose sides may or may not all be the same length.

Rhombus

A rhombus is a parallelogram whose sides are all the same length.

Its angles may or may not all be equal.

Rhomboid

A rhomboid is a parallelogram whose sides are not all the same length.

Its angles may or may not all be equal.

Trapezoid

A trapezoid is a quadrilateral which has exactly one pair of sides parallel:

Trapezium

A trapezium is a quadrilateral with no parallel sides.

Further subclassifications

Various breeds of irregular quadrilateral are unofficially and informally recognised:

Kite

A kite is an irregular quadrilateral which has both pairs of adjacent sides equal.

Dart

A dart is an irregular quadrilateral with a reflex angle.

Also known as

A quadrilateral can also (rarely) be found referred to as a tetragon.

Euclid's Definitions

In the words of Euclid:

Of quadrilateral figures, a square is that which is both equilateral and right-angled; an oblong that which is right-angled but not equilateral; a rhombus that which is equilateral but not right-angled; and a rhomboid that which has its opposite sides equal to one another but is neither equilateral nor right-angled. And let quadrilaterals other than these be called trapezia.

Linguistic Note

The word quadrilateral derives from the Latin for four sides.

Similarly, the word tetragon derives from the Greek for four sides.