The short answer to "what is a quadrilateral?" is that it is a foursided polygon. In other words it is foursided, twodimensional enclosed shape with straight sides.
Your children will learn (typically in 5th Grade) that there are many different types of quadrilaterals and they can be arranged in a hierarchy based on their properties.
Discuss the relationship between the quadrilaterals with your children. Follow the arrows down to see how certain quadrilaterals are also special types of others as they exhibit all their properties. For example, a square is a special type of rectangle. Indeed a square can be said to be a special case of all the other quadrilaterals shown in the table.
The table shows different types of special quadrilaterals arranged in a hierarchy.
Follow the arrows down (or up if your prefer) the table to view the various categories and subcategories.
Square  
A Square is a quadrilateral with equallength sides forming four rightangles.  
A square is also a special type of  ... and it is also a special type of  
Rhombus  Rectangle  
A Rhombus is a Quadrilateral in which opposite sides are parallel and all four sides are of equal length. 
A rectangle is a quadrilateral with four rightangles and opposite sides that are parallel.  
A Rhombus is also a special type of  ... and it is also
a special type of  A Rectangle is also a special type of 
Kite  Parallelogram  
A Kite is a Quadrilateral in which the sides can be grouped into two pairs of adjacent equallength sides.

A Parallelogram is a quadrilateral in which each of the two pairs of opposite sides are parallel.  
A Parallelogram is also a special type of  
Trapezoid  
A Trapezoid is a Quadrilateral in which two of the sides are parallel.  
A Kite is also a special type of  A Trapezoid is also a special type of  
Quadrilateral  
A Quadrilateral is a polygon with four sides. 
You can also try a handson activity with this worksheet to show that there are 360° in a quadrilateral.