Simply: a kind of bigger soundcard for music studios. Usually as an extern hardware device, which is connected via PCMCIA, USB or Firewire; sometimes combined with an extern keyboard. Smaller audio interfaces can be placed intern in the computer ( PCI Slot ).
However, an audio interface has the function to record (microphones or an instrument), and to play back and hear the recordings.

The input/output ports for analog audio are cinch or 1/4´´; digital connections are optical or S/P-DIF. Better devices offer AES/EBU or XLR ports, and additional features, like multi channel recording, high quality Mic Preamps, or MIDI.

Simply: It makes digital to analog, and vice versa. In detail:
The A/D conversion transforms the analog (=A) audio signal into a digital signal (=D), in order to get the music "information" into the computer ( = recording).
The D/A conversion transforms the digital audio signal into an analog signal, when it comes out of the comupter. This is necessary, when monitoring, or playing back the music.

This job of converting signals does the converter. The higher the resolution, the better the sound. However, you should give your ears a chance: Audio interfaces may differ remarkably in "the sound", although having the same resolution.
Today almost all converters offer 24 Bit and 96 KHz; but keep in mind: the music CD Format is 16 Bit and 44,1 KHz, and this format is still very popular. So in many cases recording with 16 Bit / 44,1 KHz may be convenient.

The time between A/D and D/A processing is called latency. Your computer has to work hard in order to process incoming and outgoing data, and this takes some time. Latency can be a problem, when playing an instrument. For example: you play and record the guitar (= signal goes in the computer). But what comes out of your computer (= what you hear on the studio monitors or earphones) is delayed. It´s because the signal must be transformed again, when it comes back out of the computer.
So keep an eye on latency. The better your audio interface, the smaller the latency (in milliseconds). Plugging a guitar directly into a guitar amp, of course, does not face that problem. Analog sound signal processing has no latency. Welcome to the digital world....