Like phinds said, it is easier for you to just google it. Because many others were dealing with the same problems, but i will just asume that you already did that and still did not understand everything - or I will just did it to be nice.
Anyway, let's start with your first question:
1) tangent
I've got one image for you, just so you can visualize what we are talking about [PLAIN]https://upload.wikimedia.org/wikipedia/commons/6/61/Cercle_tangente_rayon.svg.
A tangent line is just a line which gives the slope at a certain point of the function. In the image the function was a circle.
1) normal
The normal to a function is just a line perpendicular to the tangent at that point.
1) Radius of curvature, I will explain that one later. First I need to be sure that you understand what i just explained.
2) Spherical coördinates, those are the coördinates which are normally used when you want to calculate for instance the volume of a sphere, or just a part of a sphere.
I can into detail, but you need to tell me first wat exactly you want to know. Before I tell you more than you want to know and you start to get confused.
Polar coördinates are the coordinates which you will use for a 2D sphere; a circle.
But once again, tell me what you want to know.
Cylindrical coördinates are used to calculate the surface, volume and more of cylinders.
3) Taylor series: a Taylor series is a representation of a function as an infinite sum of terms that are calculated from the values of the function's derivatives at a single point.
I assume you know what a derivative is?
I know this will not help a lot, but tell me more about the things you want to know exactly.