Electronic – Darlington Emitter-follower used as a buffer

Here's the deal. The capacitor provides constant voltage at high frequencies across the BJT base-emitter + resistor combination. This causes fairly constant current through the BJT and resistor, with some high impedance Z, probably determined mostly by the BJT base resistor Rb. The FET has a high transconductance (gm = Iout / Vin), and the net gain is gm * Z. This is the voltage across the FET drain-source. The BJE emitter resistor has constant voltage across it, so there's a bias voltage added to that. The constant current allows the BJT to act as a low-impedance output buffer (=Rb/beta).

A buffer can serve many purposes in a circuit. There are other uses besides what I mention below, but these are some fairly common ones you might encounter.

One common use is as a time delay in data transmission. It can take a bit of time for a chip to recognize the change in a signal's voltage level and react to it (such as a digital HI to LO transmission). The amount of this slight propagation delay varies from one device to the next. Sometimes it is a few nano seconds, sometimes it is much more. Also, it does take some time for the signal to propagate through the copper from one part of the circuit board to another. When there are one or multiple data signals moving from one part of the circuit to another, it may be necessary to delay one so that they arrive in a specific order or at exactly the same time.

Buffers can be used as momentary data storage. Similar to the time delay, there are situations when you need a small amount of time to store a data bit such as pipelining, but this example can be a bit difficult to explain, depending upon your knowledge of computer architecture.

Another example is signal isolation. Because a buffer is essentially a signal repeater, it can be used to isolate the signal from two parts of the circuit. This is the specific use in your example, as noted by Ignacio in his answer. In this particular case, the DAC (digital to analog converter) is sensitive to loads. A load can be anything from a resistor, speaker, IC input, or even just a piece of wire. This "sensitivity to load" means the signal may change (distort) if too much or too little of a load is seen at the output of the DAC. To prevent this, a buffer chip is used to repeat the signal from the DAC output. The output of the buffer can be connected to whatever you want (within its own limitations) without affecting or distorting the output from the DAC in any way.

Lastly, a buffer can be used a signal amplifier. This is very similar to the previous example, but the reasons for using it can be different. While the output of some chip might not be extremely sensitive to loads, it may only be able to supply a few milliamps of current. In this case, you may want to use a buffer with a larger output current rating to amplifier the original signal so it can be used to drive some larger load.