Right to the point.
A Typeface is the style of lettering. Helvetica is a typeface.
A Font is the size and weight of the typeface. 18pt Helvetica italic is a font. A font can also be the files (a font family) that make the typeface display on a computer screen. In the Helvetica font family there is Standard, Bold, Italic, Bold Italic, Condensed, Compressed, Extended, Black, Inline (an outline of the characters--- like having an outline with no fill in Microsoft Word), Rounded, and several others.
So... um... why do people say "font" when they mean "typeface?" Well... I don't know. Maybe it's easier to say? Also, somehow "font" came to mean the digital file the typeface is saved in when computers started having monitors. There are some people who are really snobbish about it, but really I'm not. I'm the FontGeek. Not the TypefaceGeek. I'll sometimes use the word "font" when I mean "typeface".