In typography, a typeface is the design of the set of characters. Times New Roman is one typeface, Arial is another. There are many different categories of typefaces, including: Roman (Serif, Sans-serif, Script, Ornamental), Blackletter, Gaelic, Monospaced, Symbol.
A font is a set of characters in a particular size and typeface. For example, within the typeface Times New Roman, is a series of fonts. Times New Roman in 20 pt. is a font, and Times New Roman in 16 pt. is another font. Times New Roman in 16 pt., Bold and Italic is yet another. In other words, a font is a member of a typeface family, and designates the size and style of the typeface.