It occurs to me that we are in the midst of two generations of home recorders - those who remember records and those for whom records are a quaint historical legend. So maybe the obvious needs to be stated, as it is no longer obvious to many. It might even have entered the realm of the exotic. Makes me feel kinda old. So this is worth what it costs you - and I make no claim to accuracy. If anyone feels like correcting me, go to it and have fun. I'm easy.
Think of the idea of digital signals and analog signals as offering descriptions of the kind of basket into which audio information is tossed, and the form both the basket and the information takes.
Analog data storage devices correspond with the earliest recording technology: magnetic or vibrational signals imposed on a medium. So the earliest recordings were vibrational signals preserved in a soft medium. Folks would speak or sing into the horn of a megaphone (from the big end). In the small end was a mica disk with a steel needle stuck in the middle - the earliest microphone diaphragm, literally made of a shaving of quartz. The disk vibrated with the voice and the needle vibrated with the disk. A wax cylinder passed underneath the needle and a pattern - actually, a wave form - was carved in the wax. When the cylinder was passed underneath the needle a second time, the pattern caused the needle and the disk, or diaphragm, to move and the movement made sound. The cone of the megaphone amplified the signal and one could hear the recording. Sort of.
The wax medium was fragile and was quickly replaced by progressively stronger media - mostly hard wax and shellac, occasionally hard rubber. The recording needle was replaced with a knife edge and the recording device became known as a lathe, very similar to the lathes that have been used in woodworking and machine shops since the beginning of that technology in the late 18th and early 19th centuries. The cylinders could withstand multiple playing. That was the Edison phonograph design of about 1900; one sits in my studio today.
The next development of this technology concerned the fragile nature of shellac and hard wax / rubber cylinders. Platters were easier to store and, when supported on a felt topped turntable, were more durable than cylinders. So the technology developed records - platters in which the lathe carved an ever decreasing arc of sound waves. The "grooves" on a vinyl album (and their shellac ancestors) are in fact made of one and only one groove. If you look at these with a magnifying glass you will see waveforms carved into the medium. These waveforms cause the needle to vibrate. In the very old mechanical turntables, the vibrations caused a diaphragm to sound. In the later, electric and electronic turntables the needle is located between two very small magnets and its vibration induces an electric current exactly as a modern microphone does. The current is amplified (processed etc) and is then sent to an analog speaker - which responds to the stronger signal and causes a larger diaphragm to vibrate - the speaker cone. We hear the sound as Edison did, albeit with better fidelity.
Other analog media came and went, but some came and stayed. In the 1930s steel wire was common as a recording medium. Easily magnetized, it could cause an electric current when passed across a magnetic reading head. I remember fooling with a wire recorder in high school. It was an antique but recently so enough that it had no value - we actually recorded guitar music on the thing. It had a terrible sound, a bit like a talking dog. You don't worry so much about what it says as it's remarkable it can talk at all.
Tape became the other medium for analog signals - both reel to reel and cassette tape. Interestingly, VCRs also used the same medium to record digital signals; a VCR tape is a hybrid of digital and audio tape. The image is digital and the audio tracks in the early VCRs was an analog tape signal. ADAT tapes are digital archives, but the medium is the same as with an analog tape.
In a digital signal, sound is transformed to an electric signal exactly in the same way analog sound is transformed, and then a computer takes repeated "snapshots", or samples, of that current. The computer program interprets the signal as an "off" or an "on" event. Large numbers of these numeric documents when assembled together comprise data from which an approximation of the signal may be recreated. The sampling rate at which the approximations are captured and which appears on a standard CD is 44,100 samples per second. The bit depth for a CD is 16 bits, which may be thought of as roughly a measure of how much information (or density of signal) in each sample. Lots and lots of zeroes and ones add up to something that sounds recognizable.
Interestingly, the first digital signals (or similar kinds of signals) were developed in the 1800s with semaphore. Those early flag signals are the forerunners of today's fax machine signals - literally the same in some instances. Fax machine technology may have relied upon the development of the modern telephone line to become affordable, but its software and digital signal structure is a hundred fifty years old.
Note that when describing a digital signal I used the term "approximation." No matter how fancy the computer system is, a digital signal will by definition contain less data than an analog signal. The reason is that a digital signal is composed of samples. In between the samples is dead air space during which no data is being collected. With an analog signal there is no periodic sampling and therefore more data to work with.
While digital sampling now approaches or exceeds the ability of the human ear to distinguish most tones in the audible range, meaning at some point we won't be able to tell if its digital or not, we are not at that point yet. There remains something different about how good analog recordings sound - whether more "pure" or less so is beyond me to suggest, but clearly different. They are not the same and even now there is something nearly indefinable in an analog signal that is missing in a digital signal (along with the distortion arising from wear and design limits of the analog equipment processing the signal). People refer to it as a kind of warmth, and progressively more esoteric, spiritual or erotic terms. Basically there's more stuff in the signal and we can sense even what we can't actually hear. So we get more stuff from the analog signal, for better or worse.