A synth is a device that generates sounds. They’re often (but not always!) controlled by piano-style keyboards, but so are lots of non-synth things. For example, although a piano emulator could be a synth, they’re generally not thought of that way. A sampling keyboard is not a synth: it’s playing back recorded sounds.

Lots of synths spent their whole lives never connected to a keyboard, but might live in rack mounts and be driven by sequencers or DAWs.