I never paid much attention to faucets...until I built my first house.  Then I was overcome with the choices and technical details.  The mistakes I made were costly and purely out of ignorance.  Here is what I learned about buying a lavatory faucet.

The faucet you choose will be largely influenced by your sink.  Is your sink a drop in, undermount, or a vessel? Does it have a deck with pre-drilled holes or will you mount the faucet on the vanity top?

If your sink has no pre-drilled holes you have the flexibility of choosing any style of faucet to mount either on the vanity top or wall.  Be aware that wall mount faucets require a rough in valve which may not be included with the faucet.

If your sink has a deck with pre-drilled holes, you must know the distance between the holes for the handles. The standard distances are 4 inches (centerset), 8 inches, and single hole.  "Widespread" faucets can accommodate sinks with centers between 8 and 16 inches. In the example photo the distance between the handles is 200 mm (8 inch center).

Believe it or not the pre-drilled holes in the sink deck for the handles and spout are not standardized in terms of size.  You must measure the diameter of the holes and determine that the faucet will fit.  The technical specifications for the faucet can be downloaded from the manufacturer's web site.  In the example photo, the handle requires a 28 mm hole and the spout requires a 32 mm hole.

Spout length and height will affect the amount of splash and whether you can comfortably wash your face and hands. Measure the distance between the center of the spout to the center of the drain. This is the spout length.  Height is a matter of preference.  The higher the spout and the shallower the basin, the more splash will occur.  Take into account the angulation of the spout.  Does the water flow straight down or at an angle?

Handles are a style preference.  However, round knobs can be a problem if hands are wet, slippery or arthritic.

Better made faucets have brass or zinc die cast construction. Look for "quarter turn washerless ceramic valves".  This type of valve is more resistant to debris and hard water.

The finish is also a personal style choice.  Polished chrome will be the least expensive.  PVD finishes are very resiliant to wear, corrosion and tarnish. Some finishes will develop a patina with handling.