Ions derived from some molecules can be aromatic. In fact you can deprotonate (make an ion) no aromatic molecules and make (synthesize) ions that behave as though they are aromatic.
Probably the most famous and useful is the cyclohexadienyl anion. If you were to deprotonate (i.e. rip a proton of of cyclohexene), you will create a cycohexadienyl anion.
The cyclopentadienyl anion is aromatic. Shown above is one possible Lewis structures (resonance structures) for the cyclopentadienyl anion. There are 5 total resonance structures, each one having a negative charge on a different carbon.
Recall that for a molecule to be considered aromatic it must be cyclic, conjugated (i.e all atoms sp2) and have 4n+2 π electrons. Looking at the resonance structure above we see that the carbon with the negative charge appears to be sp3. So how can it be aromatic? Turns out the carbon with the negative charge is not sp3, but rather has significant sp2 character. Lets see why!
Let us assume the negative carbon is actually sp3 hybridized. The following 3D structures shows the cyclopentadienyl anion if this carbon is sp3. In this case the lone pair (GREEN) is in an sp3 hybrid orbital, and cannot align with the other p orbitals (Purple/yellow) of the other 4 carbons. The lone pair orbital does not have the correct shape and is pointing in the wrong direction.
If however, the carbon with the negative charge is sp2, then the lone pair electrons would have to reside in a p orbital. In this case they would perfectly align with the other 4 p orbitals and would be conjugated.
If we look at the other resonance structures, a given carbon is sp2 in 4 of its 5 resonance structures. So every C is like 4/5 or 80% sp2. Its conjugated!
There are other aromatic ions. Here are a few.