The hydrogen atom is very important for understanding electrons in atoms in general as we use the knowledge of electrons in hydrogen as the basis for our model for electrons in all the elements.
There are several key ideas that we can take away from studying the hydrogen atom that we can apply to all atoms.
The first is that the electron in a hydrogen atom cannot have any energy but instead is only found in particular fixed energy levels. This is odd. This is unlike any of the macroscopic objects we deal with in our everyday lives. This is a strange and bizarre consequence of quantum mechanics (QM). This is a critical idea for understanding atoms, molecules, and chemistry.
Second, small mass particles like electrons obey a different set of rules than the macroscopic objects of our everyday lives. They are not the same. We should not try to imagine them as being the same. However, we constantly try to classify them in ways that fit with our own macroscopic world view. However, we must always remember that they are not macroscopic particles and therefore they will never fit in with our macroscopic pictures. Electrons are simply not small particles orbiting like planets around nuclei.
Third, electrons follow the rules of quantum mechanics (QM). QM provide us two useful ideas about electrons, their energy and their "wavefunction". Energies turn out to come in discrete or "quantized" units. Wavefunctions tell us something about the "spatial distribution" of the electron. For lack of a better description "where the electron is".
Fourth, we find that the energies and wavefunctions that describe an electron in a hydrogen atom are very systematic. They can be classified by a set of "quantum numbers". These define the actual mathematical function that is the wavefunction. They effectively describe the "size and shape" of the electron's wavefunction.
Finally, we use the ideas we get from the hydrogen atom to describe all the rest of the elements. So these solutions that we obtain from QM for hydrogen are used again and again in chemistry to describe atoms and bonding.