Originally, a font (also spelled fount, at least formerly) was a physical thing: a collection of metal slugs, each bearing the reversed shape of a letter or other symbol (a glyph, in typographical parlance). You would arrange these slugs in a wooden frame, apply a layer of ink to them, and press them against a sheet of paper.
The typeface dictated the shapes of those glyphs. So you could own a font of Caslon's English Roman typeface, for example. If you wanted to print text in different sizes, you would need multiple fonts. If you wanted to print in italic as well as roman (upright), you would need another font for that, too.
As there was a finite number of slugs available, what text you could print on a single sheet was also constrained to an extent by your font(s). Modern Welsh, for example, has no letter "k": yet mediaeval Welsh used it liberally. The change came when the Bible was first printed in Welsh: the only fonts available were made for English, and didn't have enough k's. So the publisher made the decision to use c for k, and an orthographical rule was born.
Digital typography, of course, has none of those constraints: digital text can be made larger or smaller, or heavier or lighter, or slanted or not, by directly manipulating the glyph shapes; and you're not going to run out of a particular letter.
So that raises the question: what is a font in digital terms?
There appear to be two schools of thought:
1. A font is a typeface at a particular size and in a particular weight etc. So Times New Roman is a typeface, but 12pt bold italic Times New Roman is a font. This attempts to draw parallels with the physical constraints of a moveable-type font.
2. A font is, as it always was, the instantiation of a typeface. In digital terms, this means a font file: a .ttf or .otf or whatever. This may seem like a meaningless distinction, but consider: you can get different qualities of font files for the same typeface. A professional, paid-for font will (or should, at least) offer better kerning and spacing rules, better glyph coverage, etc. And if you want your text italic or bold, or particularly small or particularly large (display text), your software can almost certainly just digitally transform the shapes in your free/cheap, all-purpose font, But you will get better results with a font that has been specifically designed to be small or italic or whatever: text used for small captions, for example, is more legible with a larger x-height and less variation in stroke width than that used for body text. Adobe offers 65 separate fonts for its Minion typeface, in different combinations of italic/roman, weight (regular/medium/semibold/bold), width (regular/condensed) and size (caption/body/subhead/display).
Personally, I prefer the second definition.