EME is an acronym for Earth-Moon-Earth, or more commonly known as moon bounce. This is where we use the moon as a reflector to communicate with someone else on the planet using frequencies that would normally travel only hundreds of kilometers.
One of my earliest introductions to EME was traveling down with Frank WA1RWU (now NC1I) to visit Steve WA1FFO (K1FO SK) to help work on his modest antenna. Frank and Steve designed an EME antenna array that Frank installed at his brother Gary's (WA1ECR SK). I spent many an hour there with headphones on straining to copy bits of Morse Code (CW) to decipher a week signals being received off the moon as a reflector.
Things have improved over the decades since my initial foray in moon bounce: we have access to much better radios and electronics and with the assistance of software like WSJT, EME contacts can now be made with stations running low power and simple antennas.