The usefulness of the definitions (there are many, but let's not open a new can of worms) of bandwidth can be understood once you learn how a signal traveling through some medium or system is altered in the process.
You'll learn that not only a signal has its bandwidth, but also a system or a medium has its own bandwidth (or pass-band). The bandwidth of a system (or medium) is (roughly) the range of frequencies that the system lets pass without modification. Frequencies outside that bandwidth are altered in some way.
In particular, you'll learn that linear time-invariant (LTI) systems can be characterized in the frequency domain by a complex function called frequency response H(f). This function is important because it tells you how a sinusoidal signal is altered by the system, since a LTI system can modify both the amplitude and the phase of a sinusoidal signal.
Since the system is linear and any signal can be represented by a superposition of a (possibly infinite) number of sinusoidal signals, knowing H(f) let you compute exactly how any signal is modified by the system.
How is this babble about H(f) related to the bandwidth of the system? Because the modulus of H(f), i.e. |H(f)| gives you the information to determine the bandwidth of the system, hence the range of frequencies that can pass through the system unaltered.
So, a system (or medium) with a 2Hz bandwidth can carry any signal with smaller bandwidth, provided the signal lies entirely in the system's pass-band. If the pass-bands don't overlap, or overlap only partially, either the signal won't pass (in the former case) or will be distorted heavily.
Moreover, it can be shown that a signal's bandwidth value is related, in the time domain, to the rate of variation of a signal. In other words, a signal with a 10Hz bandwidth will vary much more slowly than a 5kHz signal.
Disclaimer: the question you made is really broad, so I had to simplify many of the subjects I touched upon and I cut some corners. Don't expect extreme rigor in the above, since it would require at least a x10 lengthier text to put all these things in a more formal framework. You can find more detail in this Wikipedia article on bandwidth.
Moeover, keep in mind that although the concept is the same, it is explained in slightly different ways depending on the specific branch of EE you are studying. For example signal theory versus control theory versus analog filter design versus network engineering: four fields where the concept is employed, but which sometimes use slightly different approaches.