A quadrilateral which has no parallel sides .
(but see definition notes below)

A trapezium is defined by the properties it does not have. It has no parallel sides. Any quadrilateral drawn at random would probably be a trapezium.
Since it has no interesting properties beyond those of a quadrilateral, it is not used much in geometry.

If the quadrilateral had one pair of parallel sides, it would be a trapezoid.
If both pairs of sides are parallel, it would be a parallelogram.

Definition notes

There is considerable confusion over the definition of 'trapezoid' and 'trapezium' due to differences in the British and US versions.
As you can see from the table below, the meanings of the two words are exactly reversed between the US and British interpretations.