Are compact sets on $\mathbb R^n$ always connected? I am unsure if compact sets on $\mathbb R^n$ are always connected. Can someone explain it to me?

As the other answers indicate, compactness and connectedness have little to do with one another. You may be thinking in terms of compact cubes, so you will have to think of more general sets.
For example, $[0,1]\cup[2,3]$ is compact but not connected, and is made up of compact intervals.
