Vinegar is often used in ceramic slurries to change the viscosity (thicken it). While there are more effective flocculants (e.g. calcium chloride, epsom salts), vinegar is popular among potters simply because it is so available.
The effect is to gel the slurry. When additions are done judiciously the degree of gel in a glaze suspension can be fine-tuned to produce a rheology that enables applying an even layer without runs (even on even dense ceramic bodies).
Of course, if there are carbonates in the glaze the acid can react with them (the thixotropy of the slurry will also be lost, theoretically because the reaction would produce CO2 and that would neutralize the acid). By-products of this reaction can have a negative effect on the suspensions ability to respond to epsom salts (if this happens calcium chloride should still work). Even if there are no carbonates, vinegar-flocculated slurries can thin out over time. When this happens just add a little more to reestablish the desired rheology.
When a slurry is very fluid (having a low specific gravity) vinegar may not be effective. In these cases epsom salts will often work (producing a more stable slurry as well).
When to use vinegar and when to use epsom salts to flocculate a slurry
Tap picture for full size and resolution
Slurries with more clay (like engobes, slips) generally respond better to epsom salts. However the extra clay also makes them more likely to go moldy, so you may need to add a few drops of Dettol to kill the bacteria (if they are stored for any length of time). Vinegar works better for glaze surries, but only if they have sufficient specific gravity. Many people like to make an epsom salts solution and add that, but if you have a good mixer you may find it more intuitive to add the crystals (which you should crush to a powder) and wait 30 seconds for the viscosity to respond.
Thixotropy Thixotropy is a property of ceramic slurries. Thixotropic suspensions flow when you want them to and then gel after sitting for a few moments. This phenomenon is helpful in getting even, drip free glaze coverage.
Rheology In ceramics, this term refers to the flow and gel properties of a glaze or body suspension (made from water and mineral powders, with possible additives, deflocculants, modifiers).