First of all, a point of osculation is...well, it's that bit inside the little red circle:
You get a point of osculation when two bits of a loopy curve have the same tangent (I'll explain tangent in a minute). The tangent in the picture above is the black vertical line. It's got a little black triangle on it, I don't know why.
All right so far? Well, okay, now the tangent stuff. Imagine you're walking along a curved path. Freeze the action at a point where both your feet are on the ground. Done that? Note the direction in which you are travelling. Then go on one more step and freeze the action again. Again note the direction in which you are travelling, which will be just a little bit different from last time because you'll have gone a bit round the curve. Well, each of those direction-lines is the tangent of the bit of the curve you're travelling along at each of those particular frozen moments. A tangent is basically a line that shows the direction the curve is heading in at any given moment.
In the picture, you can see that there's an instant where the shallow outside curve and the tight inside one are going in the same direction. So they have the same tangent. That's a point of osculation.
I hope that makes sense, but, honestly point of osculation...could anything sound more pompous?
Anyway, the other way the word osculation is annoying is that it also means kiss. Good grief. If anyone proposed osculating me I'd give them a smack on the osculater, and no mistake.
Word Not To Use Today: osculation. This word comes from the Latin ōsculārī, to kiss, from ōs, which means mouth.
For those of you doing geometry, you can also call a point of osculation a tacnode or double cusp.