Most fonts can be categorized as one of: Serif, Sans (Serif), Mono(space).
Serif fonts feature lines at the corners of characters. Sans serif fonts are more technical/mechanical/electronic. Monospaced fonts have characters all of the same width. This makes the characters align one below the other, even across several lines.
When choosing a font, apart from the visual style of the font you will have to consider two things: Will you deliver the fonts yourself (webfonts, install fonts), or use potentially installed fonts (checking for spead and providing fallback fonts are a necessity here). Also, how much Unicode does the font actually cover? Enough for your need? (Multi-lingual and -regional environment?)
Recommended, free (as in freedom) fonts: