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A web font should be legible. What makes a font legible? 
Open spaces and healthy strokes.

Things to look for in a legible font are generous x-height (lowercase letter 'x' height)
Spaces in the lowercase letters feel more generous because of larger the x-height. These spaces inside the letters are called counter forms. 

Legible fonts have open apertures. That's the opening in letters like C, A, and E (lowercase). See how it's bigger in Verdana? An open aperture creates more visual space.

Slightly loose letter spacing also promotes legibility. The text in Verdana is clear to read. Generous bowls also promote legibility. The bowl is the shape of the rounded forms in the lowercase letters. If a bowl is too narrow, the counter form is too small, if it's too round, the counter form gets too big. Look for a bowl that falls nicely in the middle. Also related to bowls and counter forms or shoulders. Shoulders are where the curved line meets the stem in letters like H, D and M.

Ascenders are strokes that extend above the main line. Descenders are strokes that extend below the baseline. Slightly longer ascenders and descenders or having serifs on the ascenders and descenders can help make the shapes of the words more recognizable.

Serifs are the little strokes coming off the ends of stems in some fonts. We call these serif fonts. The fonts without serifs we call sans-serif fonts. Does a legible font need serifs? One common misconception is that text should be set in a serif font because it's easier to read and sans- serifs should be used for headlines. This is not always true. Both serif and sans- serif fonts can be legible or lack legibility. Here Verdana is more legible for text than Times New Roman.

Strokes and spaces are far more important than serifs when it comes to how easy a font is to read.

A legible font has discernible terminals. Terminals are the ends of strokes in letters like A, R, and F. Some terminals have a ball or pen-formed shape. Other terminals don't have any extra shape to them. While you don't want a font with a crazy shape like blood or thorns, having a bit of shape can help readers differentiate between the letters in text.

Finally, legible fonts have generous strokes. If strokes are too thin, they get lost. If they are too thick, you run the risk of losing space in the letters