Analog vs. Digital
Before going too much further, we should talk a bit about what a signal actually is, electronic signals specifically (as opposed to traffic signals, albums by the ultimate power-trio, or a general means for communication). The signals we're talking about are time-varying "quantities" which convey some sort of information. In electrical engineering the quantity that's time-varying is usually voltage (if not that, then usually current). So when we talk about signals, just think of them as a voltage that's changing over time.
Signals are passed between devices in order to send and receive information, which might be video, audio, or some sort of encoded data. Usually the signals are transmitted through wires, but they could also pass through the air via radio frequency (RF) waves. Audio signals, for example might be transferred between your computer's audio card and speakers, while data signals might be passed through the air between a tablet and a WiFi router.
Analog Signal Graphs
Because a signal varies over time, it's helpful to plot it on a graph where time is plotted on the horizontal, x-axis, and voltage on the vertical, y-axis. Looking at a graph of a signal is usually the easiest way to identify if it's analog or digital; a time-versus-voltage graph of an analog signal should be smooth and continuous.
While these signals may be limited to a range of maximum and minimum values, there are still an infinite number of possible values within that range. For example, the analog voltage coming out of your wall socket might be clamped between -120V and +120V, but, as you increase the resolution more and more, you discover an infinite number of values that the signal can actually be (like 64.4V, 64.42V, 64.424V, and infinite, increasingly precise values).
Example Analog Signals
Video and audio transmissions are often transferred or recorded using analog signals. The composite video coming out of an old RCA jack, for example, is a coded analog signal usually ranging between 0 and 1.073V. Tiny changes in the signal have a huge effect on the color or location of the video.
Pure audio signals are also analog. The signal that comes out of a microphone is full of analog frequencies and harmonics, which combine to make beautiful music.