Local tangent plane

Imagine you are drawing on a big balloon with a marker. If you draw a line on the balloon, it might look like a straight line to you, but it's not actually straight because the balloon is curved.

Now imagine you zoom in really close to the line you drew. If you look really closely, you might notice that the line starts to look more and more like a straight line - this is because you are getting closer and closer to it.

If you kept zooming in infinitely close to the line, at some point it would basically look like a straight line. That's what a tangent line is - a line that is really close to a curved line at one specific point.

Now let's talk about a tangent plane. Instead of just looking at a line, let's say you're looking at a curved surface - like the outside of a ball. If you zoom in close enough to a point on the surface of the ball, you'll see that it starts to look more and more like a flat surface.

The tangent plane is a flat surface that is really close to the curved surface - it's kind of like a little piece of paper that you are holding up against the ball. The tangent plane is only flat at one point on the curved surface - if you move even a little bit away from that point, the tangent plane won't be flat anymore.

The local tangent plane is just the name for the specific flat surface that is really close to the curved surface at one specific point. It's like taking a picture of the flat surface that is touching the curved surface at that one point.