Simple cyclic hydrocarbons with 5, 4, or 3 carbons are called, respectively, "cyclopentane", "cyclobutane", or "cyclopropane".
Usually, we draw them with linear bonds, but bonds are an artifact of the mind — only orbitals, atom positions, and potential energy are real — so we might as well curve the bonds, to emphasise their circular structure:
The trend I've shown goes from 5 to 4 to 3. The obvious next step is to continue to 2: "cycloethane".
Straighten out the curved bonds and you get what looks like a double bond. Indeed, little sense is to be found in cycloethane except by a double bond. Cycloethane is ethene. Double bonds are a special case of rings.
In a sense, this is misleading. Double bonds combine fundamentally distinct σ and π bonds, but the interpretation as cycloethane treats the two parts as equivalent. I have no response to this technicality beyond acknowledging it.
Hydrogen deficiency (HD), or unsaturation index, is a number associated with an organic molecule which relates the numerical formula to the chemical structure. On the structural side, it is conventionally said that HD equals the number of rings, plus the number of double bonds, plus twice the number of triple bonds. By equating double bonds (and triple bonds, analogously) to rings, we can simplify this rule by saying that HD equals the number of rings, and nothing more.
You may object that, when combined with the interpretation of double and triple bonds, this alternative form is no simpler. In that case, view it as a mnemonic.
You could go further with this concept and name molecules in terms of rings, eschewing all references to double or triple bonds. That process gets tedious quickly.