A convex polygon is defined as a polygon with all its interior angles less than 180°. This means that all the vertices of the polygon will point outwards, away from the interior of the shape. Think of it as a 'bulging' polygon. Note that a triangle (3-gon) is always convex.
A convex polygon is the opposite of a concave polygon. See Concave Polygon.
In the figure above, drag any of the vertices around with the mouse. Take note of what it takes to make the polygon either convex or concave. Also change the number of sides.
All the diagonals of a convex polygon lie entirely inside the polygon. See figure on the left. (In a concave polygon, some diagonals will lie outside the polygon).
The area of an irregular convex polygon can be found by dividing it into triangles and summing the triangle's areas. See Area of an Irregular Polygon
Regular Polygons are always convex by definition. See Regular Polygon Definition. In the figure at the top of the page, click on "make regular" to force the polygon to always be a regular polygon. You will see then that, no matter what you do, it will remain convex.