FFmpeg is a free, open-source project providing command-line tools and libraries for decoding, encoding, transcoding, muxing, filtering and inspecting essentially every audio and video format in use.
Its practical significance is how much of the ecosystem sits on top of it. Desktop applications, browser-based compressors compiled to WebAssembly, server-side media pipelines and the encoding stages of large platforms are, very often, FFmpeg with an interface. When a web tool says it uses "advanced compression algorithms", the algorithm is usually libx264 — FFmpeg's H.264 encoder — with settings someone chose.
This is useful to know for two reasons.
It sets a floor on what is possible. If a tool cannot do something FFmpeg cannot do, that is physics rather than a product decision. Fitting a 45-minute 4K video into 8 MB at watchable quality is not a feature anyone is withholding.
It makes claims checkable. A quality comparison stated in terms of a codec, an encoder, a preset and a CRF can be reproduced by anyone with FFmpeg installed. One stated in terms of a brand name cannot. The CRF sweep this site publishes was produced with local FFmpeg using libx264 at preset medium, recorded with the source file hashes, for exactly that reason.
You do not need to use it directly. But a tool whose behaviour cannot be described in FFmpeg's terms is a tool that is not telling you what it does.