First, try the problem Circles in Quadrilaterals to familiarise yourself with the properties of tangential quadrilaterals.

A bicentric quadrilateral is both tangential and cyclic. In other words, it is possible to draw a circle inside it which touches all four sides, and also to draw another circle around it which passes through all four vertices. (The two circles do not necessarily have the same centre!)

Here is a picture of a bicentric quadrilateral.
There is a formula for finding the area $A$ of a bicentric quadrilateral:
$$A = \sqrt{abcd}$$
where $a,b,c$ and $d$ are the lengths of the four sides.