What exactly is a connected componnent in an image?
Let S represent a subset of pixels in an image.

This is typically a set of pixels with the same value, for example a group of pixels with the value 1, surrounded by pixels with a value 0.

Two pixels p and q are said to be connected in S if there exists a path between them consisting entirely of pixels in S.

A path is a set of steps that take you from pixel p to pixel q, and where each step is a jump from a pixel to a neighboring pixel. Thus, two pixels are connected if you can travel from neighbor to neighbor starting at p until you reach q.

If it only has one component, and that component is connected, then S is called a connected set.

If all pixels in the set S are connected to all other pixels through a path, then S is a connected component.
Thus a connected component is the very intuitive concept of a continuous region of equal-valued pixels.