Imagine that to each point of 3-dimensional space we attach a little frame consisting of 3 orthonormal vectors. Suppose we could take a trip through this space as a tiny particle, and that all we could see at any given time was the little frame attached to our current location. If the frames were assigned in a smooth enough fashion, as we flew along our course, the sequence of frames we were passing through would seem to rotate smoothly.
Now this rotation function--which is the change we'd observe in these little frames over time--could be written down for any trip a particle takes through space. More generally, it could be written down for any smooth 2-dimensional surface lying in 3-space, or even for all of 3-space itself.
You could write this function down in terms of, say, the standard Cartesian 3-space coordinates. However a clever way of writing down this function without even needing to conjure up some absolute frame of reference is to express the change in each frame vector at a point in terms of the frame vectors themselves. That way, you wouldn't need to assume that space was Euclidean overall, just perhaps that it is locally Euclidean. This preserves a certain amount of generality that turns out to be worth it, for space (or space-time) doesn't have to be a big Euclidean grid, and in fact probably isn't.
Now what is all this good for? Well let's say the frames you assigned to 3-space weren't just arbitrary. Suppose you attached the three frame vectors to a curve such that one is always tangent to the curve in the direction of travel, another is perpendicular to the curve in the direction of its curvature, and the third is perpendicular to them both. This is called a Frenet frame, and these frame vectors are the tangent, normal, and binormal vectors respectively. Together with their derivatives they completely characterize the shape of the curve: how much it curves, how much it twists, and in what directions.
In general when you write down how a frame field changes over an entire space you get a connection form. As with curves, the connection form for a surface tells you everything you need to know about its shape.