Analog electronics use a continuous signal. A continuous signal is where if you had a paper of infinite length, you could draw the signal without ever lifting your hand. The signal (as far as electronics are concerned) essentially represents the voltage or current at that particular point in time. Since the real world is analog, it was probably easier to make electronics analog at first. But it's losing ground as the dominant form of signaling due to everything also capable of being digitized. However, at the end of the day, what you get out is analog, because the world is analog. The defining quality of an analog signal is that it's easy to amplify, attenuate, and filter out a meaningful signal, even if the signal appears to look like garbage. As an example, when tuning into a radio station, it's first static, then static and the radio station, and then finally the radio station as the tuner can filter out the proper frequency from the "noisy" signal. The radio waves that also travel in the air are very low power, but the correct signal is filtered out before being amplified and making its way to the speakers. Another advantage of this is that when the signals degrade, it's a gradual degradation. The problems with analog signals, in regards to electronics, is that they consume more power as components are rarely completely off, and the maximum value range of their variables is limited by the quality of its components and the amount of power they can safely handle; an analog computer pushed beyond its tolerances can be a real world example of both Tim Taylor Technology and Explosive Overclocking. If you're designing or analyzing analog circuits, it also makes for some Mind Screw math such as Fourier series and Laplace transforms. Another issue is that due to the random nature of the universe, it's very hard, if not impossible, to manipulate and maintain an analogue signal perfectly: gears skip, belts slip, pipes leak, and electrical currents fluctuate. This inherent randomness can actually be an advantage in some cases, however, as it allows for rapid, naturalistic "fuzzy math" modeling of inherently chaotic and vaguely defined situations. A common historical usage for electronic analogue computers has been in modeling air circulation and liquid flow through complex pipe systems, for instance. They have also been used to rapidly generate accurate-enough solutions for certain differential equations which are extremely difficult or tedious to solve using digital computation or traditional precision calculation. In short, analogue electronics are less efficient and harder to program, but are also more robust and tolerant of errors, and model the world more realistically. In trope terms, an analogue computer embraces "Don't Think, Feel".
Digital electronics use a discrete signal, that is, there are hard values with nothing in between them. You couldn't take a pencil and draw a digital signal without lifting it up. Despite what the picture on the right shows, vertical lines are just representations. Though in the real world, the signal is made of a sum of sine waves such that it creates a tiny sine wave that swings wildly periodically. This is known as a Fourier series. A digital signal can either be a binary signal, in which case an "on or off" state, or a series of defined levels. In a binary digital system, typically the high voltage is usually 1.5V, 3.3V, or 5V with the low voltage being 0V or the negative of the high voltage. The defining characteristic is that digital signal can be copied perfectly. Small bits of noise also don't kill the signal, as long as the value being represented is within tolerance. In regards to electronics, digital signals require less average power, since they can be in a state that's fully off or mostly off. The main problem with digital signals is that if part of the signal is trashed, then the entire signal has to be thrown away unless something to correct it is available. This is akin to handwriting, where if someone writes a letter sloppily, forgets a letter, or misspells the word, you may not be able to make out what the word really is. And if you're in a part of the world that has digital TV, you can find it very annoying that poor reception means the channel cuts out completely, rather than just getting staticky like in an analog system. A digital computer, in short, requires the employment of a Abstract Scale to function note and is inherently unstable due to the efficient approximations it has to make, but makes up for this with versatility and simplicity.