An audio interface (sometimes called a sound card) gets audio from the outside world into your computer. Many audio interfaces (especially low-end ones) contain analog-to-digital (AD) and digital-to-analog (DA) converters, but some require external conversion. Many computer motherboards have a built-in audio interface, but these generally are built from the cheapest possible components; hence their poor overall quality.
A preamp boosts a low-level signal to a higher level. Most often they are used to connect a microphone's output to the "line" level needed for a typical line input.