In modern usage, with the advent of digital typography, “font” is frequently synonymous with “typeface”, although the two terms do not necessarily mean the same thing. In particular, the use of “vector” or “outline” fonts means that different sizes of a typeface can be dynamically generated from one design.
Each style may still be in a separate “font file”—for instance, the typeface “Bulmer” may include the fonts “Bulmer roman“, “Bulmer italic“, “Bulmer bold” and “Bulmer extended”—but the term “font” might be applied either to one of these alone or to the whole typeface. Source: wikipedia.org
The Infographic below is a good sampling of 5 popular types of font styles. Source from Visually.