In the days of metal type, a “typeface” was a particular design for an alphabet, and a “font” was the manifestation of that design as a single batch of printing types. A printer who wanted to use the Franklin Gothic typeface would order a set of fonts for each size in which they were needed: perhaps one font of 24 point Franklin Gothic, another font of 18 point, and three or four fonts of 12-point, just to make sure there was always an ample supply of letters. (Like the rubber stamp kit you had as a kid, fonts contained only a fixed number of each letter, which were provided in proportion to their frequency.)
Since alphabets are now both designed and recorded digitally, they can be resized at will, and never run out of letters. As a result, the words “typeface” and “font” have come to be used interchangeably. You can sometimes detect vestiges of the old practice, when people use “typeface” to mean a particular design, and “font” to refer to the digital file that embodies that design. One often speaks of “choosing a typeface,” but “installing a font on your computer.”
People who design typefaces are “type designers” or “typeface designers.” A “typographer” isn’t a type designer, but rather someone who works with typefaces, such as a book designer.
Finally, neither “font” nor “typeface” are collective nouns: one doesn’t “design typeface,” one “designs a typeface.” To say that someone “uses font” is akin to saying that they have dog, that they love cookie, or that they’re going to see Star War. Support your local type designer by remembering to include that final S, which was such a challenge to draw!