Analog technology, as the name suggests, refers to technologies that are analogous to the process occurring behind the technology. Consider, for example, the process of converting a piece of film into a developed photograph. The film negative is used to print a representation of the scene you photographed. The developed photograph, therefore, is an analogy of the image you captured.
Similarly, the process of taking an audio signal (the human voice), and translating it into pulses over the telephone wire, and translating back to an audio signal (the same human voice) is an analogous process.
All analog technologies are limited by their physicality. How many songs you can record on an audio cassette depends on how long the plastic reel of the cassette is. How many transcontinental telephone calls you can make at the same time is limited by the number of copper and fibre optic lines laid beneath the oceans.