How to interpolate sequential points to obtain functions and/or vectors?

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I would like to know how I can interpolate a sequence (time) of points in order to obtain curves as some kind of mathematical functions.

Unfortunately math is not my area so I don't really know the name of what I'm looking for... so I mock-up a couple of pictures :)

Example of sequential points:

The information related to those points are:

  • coordinates $(x, y)$
  • creation time

example sequential points

What I can imagine as result might look like this:

A sequence of functions/point/vectors generated by matching the points

example final result

I would like to know if there is/are mathematical functions to perform this kind of operation. I guess, I'm looking for approximate interpolation, and to eventually compute a value that express the interpolation accuracy.