A convex set that has the property where at + (1-t)b is also in the set for a,b in the set and t is in [0,1]. This assumes that a and b are quantities in some vector space (which has addition and scalar multiplication) and if it isn't a vector space, then you need to make it one.
A convex function is convex if the mapping of the function has the property f(t*a + (1-t)*b) <= t*f(a) + (1-t)f(b).
Basically the intuitive way to picture a convex set is that it has no "bumps". If it had a bump the you could draw a line from the bump to another point on the boundary of the set and the line would go through some points not contained in the set.