What is usually most noticeable about a varnished piece is the crispness of its illustrations and the depth and saturation of its colours. Images may stand out or recede.
Varnishes can separate images, create emphasis, contrast, interest and surprises.
Varnish on an unprinted page can create its own image or effect.
On a single colour black it will increase depth and gloss.
The highest ink gloss on a matt job is achieved by printing gloss inks on a gloss stock and matt varnishing the text and background.

A varnish is an unpigmented ink film applied to the unprinted sheet or over varnished onto ink already applied. Varnish may be an overall application or spotted only onto the inked areas, (Spot varnish).
Varnish may be gloss, matt or satin.
Varnishes can be used to highlight or differentiate parts of a job, to protect the ink film from scuffing and marking or for increased or decreased gloss on a job.

Most common terms are

Wet trap.
A machine varnish applied to a wet ink film often on the 5th or 6th unit on a multi colour sheet fed press. Economical but it gives the lowest level of protection and increase in gloss. Varnish can be either spot or overall.
Dry trap.
A machine varnish is applied over a dried ink film. This entails another pass through the press (one colour only required), but does give better protection and higher gloss.
UV varnishes.
These film or spot varnishes offer very high protection levels and extremely glossy results but the process has considerable production problems and environmental issues.
IR varnishes (aqueous).
IR, (water based) offers a finish not far below that of UV. Multi colour sheet fed presses with inline IR coaters are readily accessible.
Film laminates, (Cello glaze)
Film laminates are not really varnishes as a laminated film is applied to the job by a specialized trade house. It does however offer similar features to an overall varnish. These films can be gloss or matt and provide the highest protection. Laminated film cannot be applied as a spot process except by clear foil stamp printing.

Watch out for.
Compatibility of the unprinted stock to the varnish or coating. Pre testing is recommended if the printer has not previously used the same varnish/paper combination.
Orange peel effect on varnish where incompatible ink/varnish combination has been used.
Unwanted higher gloss created by an overall varnish on printed areas compared to unprinted areas.
A slight yellowing of the job when machine varnish is applied, the yellowing increases with age.
If the varnished job has to be subsequently overprinted, ensure a suitable varnish is selected . Some varnishes use wax to improve gloss and slip, the wax can cause ink rejection.
Don't over varnish areas where glue is to be applied; the glue may not be able to key.
Varnish will hold set off spray, so excess spray will leave a gritty feel.
Varnish is susceptible to gas ghosting. If possible finish the print and varnish side first before backing up.

Varnish on most uncoated paper will be absorbed into the surface and is more suitable for sealing than for highlighting.