A spectrum is a band of frequencies that are emitted by an object. It is the range of frequencies or wavelengths of electromagnetic radiation present in a given sample.
The light that we receive from the Sun has a vast range of frequencies (including infrared, red, violet, green, blue, yellow , ultra violet, ...). These wavelengths ordered in increasing or decreasing order will be called the spectrum of Sun light.
Hydrogen gas emits radiation when the atoms are given some energies through heating or by light falling on them. The ordered sequence of all the wavelengths (or frequencies) emitted by Hydrogen gas molecules is called the Hydrogen spectrum.