How is a digital signal turned into a analogue signal?

Yes, Randy P is onto it. An DAC (diigital to analog converter) is given numbers by a computer very quickly. These numbers are translated to voltages. DACs come in various resolutions, typically 12 bit or 16 bit, meaning they have 4096 or 65536 discrete voltages. Enough voltages changed quickly enough makes a smooth looking waveform. Some microcontrollers have ADCs and DACs built right into the microcontroller which allows the data to be read from or written to memory without ANY work by the microcomputer. There are also ways simple ways to make less accurate ADCs. For example, simply measure how long it takes a resitor/capaictor circuit to charge. This is often good enough if the accuracy need mostly be relatively accurate.

The basic technology is called (surprise!) a digital-analog converter (DAC) or an analog-digital converter (ADC).

I'm not 100% sure how these two technologies work. I think the DAC works by adding voltages together. For instance the first bit could decide whether to add 0.1 V or not. The 2nd bit could decide whether to add 0.2 V or not. The 3rd bit would control 0.4 V or not, etc.

The resulting voltage is "quantized", like a picture made of pixels. It steps between discrete values. In my example, you'd never get 0.25 V. So filters are used to smooth it out.

I think ADCs work by comparing voltages. Each bit corresponds to whether the input voltage is or isn't more than a reference voltage.

Well you could use a component called Schmitt Inverter which inverts the signals