"Type" in Middle English meant symbol, figure, emblem.
Types are individual letters or characters used for printing, traditionally cast or engraved blocks. Nowadays more commonly known as sorts or type sorts.
Type is text printed with such blocks, or imitating its characteristics, such as on a computer screen.
Typography is the art or process of setting (typesetting), arranging and printing type. It is also used to refer to the appearance and style of typeset matter ("I like that typography"). Typography may encompass the actual designing of characters and typefaces, but it's more precise to call that artform type design.
A font is a set of type sorts of unified design: typically an alphabet plus all its accessory characters, of a single style and weight (and a single size, in the case of metal types). e.g. Neue Helvetica 36 Thin Italic.
A typeface is one or more fonts of unified design: it may include fonts of multiple weights, roman/italic, stylistic variants and even different 'optical sizes', used to replicate the subtle design modifications in different sized metal fonts. When many such variants are offered, as in the case of Helvetica, it is often referred to as a type family.
(Answered with a little help from wiktionary and Robert Bringhurst.)