A synthesizer, often abbreviated as "synth," is a musical instrument or electronic device designed for sound synthesis. It is an instrument that generates and manipulates sound through the application of various synthesis techniques.
Where synthesis is the broader concept of creating and manipulating sound, synthesizers are the "hardware" that synthesis techniques are applied upon.
A synthesizer typically comprises various components, including oscillators (sound generators), filters, amplifiers, envelope generators (ADSR), modulators, and a keyboard or controller interface. These components work together to produce and shape sound.
Synthesizers come in many forms, ranging from hardware synthesizers with physical knobs and keys to software synthesizers (virtual instruments) that run on computers. There are also hybrid synthesizers that combine analog and digital technologies.
A companion course to the fabulous Learning Music
Circles Sines and Signals - Introduction
From one of Ableton's engineers Jack Schaedler is an introductory resource for digital signal processing (DSP).
Serum is the most popular wavetable synthesizer plugin. It is full-featured, integrated into your DAW, and highly visual.
Xfer Serum Tutorials w/ Steve Duda - YouTube
This series of tutorials is from the creator of Serum, Steve Duda.
Cabbage Audio has ove 200! free and open soruce plugins. For people interested in developing their own VSTs or AUs, Cabbage also serves as an audio development framework. Their Documentation is comprehensive and easy-to-understand.