help with visualizing convex hull and polytope

I am currently taking a course in optimizations and have some problem with convex analysis.

The problem is I do not see the difference of a convex hull and polytope.

The definition of a convex hull is:

conv(V) := {a1*v1+....+ak*vk | a1,....,ak >= 0; a1+...+ak = 1}

so this gives a set. I seems i cannot see what this definition gives me (for example in 2D). for example: is the convex hull of 3 vertices in R^2 the set of points along the line of the convex hull (the convex hull itself) or is the convex hull of these 3 vertices all points enclosed by this hull?

Then how is it different from a polytope. Is a set a polytope if the set itself is the convex hull? meaning the set is convex.

Hope I am stating my problem clear enough.