EPS, short for Encapsulated PostScript, is a file format for vector graphics used mostly in professional print. Because it stores artwork as math, lines and curves rather than a grid of pixels, the same EPS logo can print razor sharp on a pen or on the side of a truck. The file does not contain an image so much as the instructions to draw one, which is why size never costs it any quality.
A helpful way to picture it: a vector file is like a recipe, while a pixel file is like a photo of the finished dish. The recipe can be cooked at any size and still come out right, whereas blowing up the photo just makes it grainy. That is why printers ask for an EPS of your logo and why it usually carries CMYK color, the standard for ink rather than screens. The format dates back to the late 1980s, which is part of its problem today. It is old, heavy, and tied to print software.
For the web, EPS is rarely used directly. Browsers cannot render it. Designers prefer formats like SVG for vectors or a PNG for raster images, so someone exports the EPS into something a browser can actually show. Think of EPS as the master copy in the vault, not the file you ship to a site.
At TopDevs we keep a clean EPS or SVG master of every client logo, so we can hand off a print-ready file and a responsive logo for screens from the same source.