Digital = number-ish (digit)
Analog = Like, or resemblance (sometimes continuous)
Quantum = Existing only in a certain subsets of numeric states, and never in between these states.
In the context of recording:
Analog means that the recording is continuous (down to the level of the recording media anyway. A groove in a vinyl disk for example, or a variable strength magnetic field in a metal impregnated tape.) Digital recordings, by contrast, store only one value per a given interval, in other words digital recordings are "chunky".
One way to visualize it; Say you record a sound wave on a piece of graph paper using a pen. The analog recording would be a continuous curve, while the digital recording would look more like a line of steps going up and down that approximates the analog version.
(As an aside; in practice there is no such thing as an analog recording, all recordings are digital if you look closely enough.)
The idea of a digital universe (as I understand it) speculates that at a small enough scale all measurements are in discreet units. This conjecture comes from quantum physics in which time and space are both quantized below a certain boundary (Planck length, Planck time, named for Max Planck)
Is the universe like this? The essential question is whether or not space and time come only in quanta, or if that's only a property of energy/mater. The math indicates that space-time is quantized. Unfortunately for the analog universe though quantum physics' mathematical prophecies have a habit of being confirmed as reality.
This quantum nature that quantum physics requires conflicts with relativity, another very strong well supported theory, causing aggravation for the Grand Unified Theory (GUT) seeking types.