These terms refer to linguistics and in particular, phonetics. It refers to a sound made by the back of the tongue against the soft palate (or velum). Examples in English include k, g, and ng (as in "sing"). If the soft palate is raised to block the nasal tract, this is what is known as a "velic closure".
A "velaric closure" is different in that velaric sounds utilise the air generated by a velic closure. The back of the tongue is raised against the velum, as before, but in addition, the lips or front part of the tongue create a clicking noise. We use this in English in "tut tut" but the click is very important in some languages such as Xhosa and Zulu.
In a velic closure, the velum moves up and comes into contact with the rear wall of the pharynx thereby blocking the inflow of air to the nasal cavity. By preventing air escape via the nasal cavity and permitting air escape via the oral cavity alone, oral sounds are produced. This type of closure produces consonant sounds like p, w, v, and x.
In a velar closure, there is a combination of two actions. The velic closure takes place in addition to the back of the tongue lifting and coming into contact with the velum. Consequently, air passage through the nasal and oral cavities is hindered and pressure in the mouth reduced. This type of closure results in click sounds.
According to Dr. Nadja Nesselhauf, velic closure refers to the manner of articulation while velar closure refers to the place of articulation.