A trapezoid is a quadrilateral with at least one pair of parallel sides. No other features matter. (In English-speaking countries outside of North America, the equivalent term is trapezium.)
The parallel sides may be vertical , horizontal , or slanting . In fact, by the definition, even this is a trapezoid because it has "at least one pair of parallel sides" (and no other features matter), as are . In these figures, the other two sides are parallel, too and so they meet not only the requirements for being a trapezoid (quadrilateral with at least one pair of parallel sides) but also the requirements for being a parallelogram.
The definition given above is the one that is accepted within the mathematics community and, increasingly, in the education community. Many sources related to K-12 education have historically restricted the definition of trapezoid to require exactly one pair of parallel sides. This narrower view excludes parallelograms as a subset of trapezoids, and leaves only the figures like , , and . This narrower definition treats trapezoids as if they are triangles with "one vertex cut off parallel to the opposite side." Even with the restricted definition, it is important for students to see non-standard examples -- asymmetric like the green and tan examples, and in non-"level" orientations like the red example -- so that the image that they build focuses on the essential feature: the pair of parallel sides.
For the area of a trapezoid, see that article.
Parallelograms with special features, like right angles or all congruent sides (or both), are given their own distinctive names: rectangle, rhombus, and square. The only special feature of a trapezoid that is awarded its own distinctive name is the second pair of parallel sides, which makes the special trapezoid a parallelogram. When two sides (other than the bases) are the same length, the trapezoid is referred to as isosceles (an "isosceles trapezoid"), just as triangles with two equal-length sides (other than the base) are called isosceles triangles. No other distinctive names are used for trapezoids with special features (like right angles or three congruent sides).
What's in a word?
The suffix -oid suggests being "like" something, without being quite the same: a spheroid is sphere-like, but not necessarily a perfect sphere; a humanoid is like a human, but not human; and a trapezoid is trapeze-shaped, but not a trapeze. The modern meaning of trapeze suggests a circus swing (that is often trapezoidal in shape, the seat being parallel to the bar from which the trapeze hangs), but trapeze originally meant "table," from tra ('four' as in tetra-) pez ('leg' or 'foot' which we more often see as ped as in pedal or pedestrian).