This is a bit of toy model, but it will likely answer your question.
All of this comes originally from spectroscopy. When you heat a substance it begins to give off light. When you pass the resulting light through a prism, you see a series of thin lines with a characteristic distribution that's unique to that element.
It wasn't known at the time, of course, but this was due to the energy levels you refer to in your question. Specifically, each of the lines in the spectrum corresponds to the light given off when an electron transitions between two of the possible levels. Since those levels are defined largely by the structure of the nucleus, each element has its own structure.
A wrinkle was noted almost from the start: when the substance was placed in a strong magnetic field, the spectrum changed - subtly. The overall structure was the same, but now where there used to be one line there were two, very closely spaced. The spectrum now had a "fine structure". Over time we learned that this was due to the effects of the magnetic field on the energy levels, causing there to be two possible energy states depending on the spin of the electrons. There's two because there's two spins.
The hyperfine structure is similar in concept, but due to a different mechanism. In this case it is not an external field that causes the lines to split, but the internal state of the nucleus itself. Depending on the energy state of the nucleus, the internal components take on different "arrangements" (for lack of a better term) and this causes slight changes in the allowed energy levels for the electrons. In comparison to the fine structure's two levels, hyperfine had four levels, and required the introduction of a quadrupole "spin" to explain it.
As others have stated, this is a very weak effect compared to the fine structure, which is one of the reasons it took Michaelson to notice it (the guy was a god of spectroscopy).
Now I'll bet the original thing that prompted your question was this:
The second is the duration of 9 192 631 770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the cesium 133 atom.
So what they are saying is this: at the lowest possible electron energy level in cesium 133, the "ground state", you don't actually have a single line, but two VERY closely spaced ones. When an electron moves from one to the other it will give off a very specific frequency of light.
Now you could probably do the same thing with any other two levels, but those are too easy to change. For instance, if you defined the second based on two normal levels of the cesium 133 atom, say the ground and first excited state, well then the frequency changes if there's a magnet nearby. By defining the standard on the hyperfine levels, they're using a state that is much more resistant to environmental effects.