That mess of knobs and buttons has been around since the '50s — longer than the more compact '80s synths: en.wikipedia.org/wiki/Modular_synthesizer Because of their size they are usually considered studio gear and not stage gear, which may also explain why the more compact synths were more visible earlier, because you rarely got to look into studios then compared to now.

To answer your question: A synthesizer (when talking about sound) is an instrument that generates sound by creating waveforms and possibly combining them in different ways to achieve different sounds. Typically they come with filters and envelopes, that further affect the resulting sound.