What are the differences and similarities between an aliphatic and an alicyclic compound?
Aromatic compounds (wiki) (about.com) (study.com) (chemguide) (khan)

Aromatic compounds are compounds that contain one or more aromatic rings.

Each aromatic ring, in its unsubstituted form, has the chemical formula $\ce{C_nH_n}$ (e.g. benzene), or with one or more of the $\ce{CH}$ groups replaced with an atom from Group V or Group VI, usually $\ce{N}$ (e.g. pyridine) or $\ce{S}$ (thiophene) or $\ce{O}$ (furan). They are usually stabilized by resonance.

Aliphatic compounds (wiki) (about.com) (study.com)

Aliphatic compounds are hydrocarbons (wiki, compounds containing only $\ce{C}$ and $\ce{H}$) which are not aromatic.

An example would be hexane (wiki).

Alicyclic compounds (wiki)

Alicyclic compounds are aliphatic compounds that contain one or more rings.

An example would be cyclohexene (wiki).

Gallery

Benzene, aromatic (courtesy Introduction to Organic Chemistry):



Furan, aromatic (courtesy Wikipedia):



Cyclohexane, alicyclic (courtesy Jarad Chemicals):



Cyclohexene, alicyclic (courtesy Exporters India):