Definition
A codec is a technical agreement for shrinking large media data, like video and sound, so it can be sent and stored, and then unpacking it so you can watch or hear it again. Most video and music codecs throw away details your eyes and ears can hardly notice, so the unpacked result isn't identical to the original, just very close.
Shrink it and unpack it, like a comforter storage bag
When you shoot a short video on your phone, the camera snaps dozens of pictures every second without a break. What if you saved those pictures exactly as they are, with no compression? Just a few minutes of video would take up tens of gigabytes, and your phone's storage would fill up in no time.
Sending that much data over the internet would be even harder. That's where a size-shrinking technology comes in. Before the data is sent or stored, it gets packed down small. Later, when it's shown on screen, it gets unpacked into something very close to the original.
The technology that does this job is called a codec. The name even blends two words: 'coder,' which compresses information, and 'decoder,' which unpacks it. Every time you watch YouTube or Netflix, your device is busy doing unpacking the data in real time and drawing it on your screen.
How does it shrink the size and still keep the quality?
The secret to cutting data size so dramatically is 'leaving out the repeated parts.' Picture a news anchor talking at the desk. The anchor's mouth and eyes are moving constantly, but the blue studio backdrop behind them barely changes for minutes.
A smart codec doesn't save that unchanging backdrop again and again. It reuses the backdrop from the first frame, and in the next frame it records only the changes in the mouth and eyes that actually moved. Just trimming away repeated information like this shrinks the data a lot.
Codecs also take advantage of the quirks and limits of human eyes. Our eyes are very sensitive to changes in brightness, but not so good at telling tiny color differences or fine detail in dark corners. A codec boldly erases information you wouldn't notice anyway, so the picture still looks sharp while the data becomes light as a feather.
A bit more precisely: file extension vs. codec
People often see letters like MP4 or AVI at the end of a video file name and assume those are codecs. But to be more precise, they aren't codecs. They are file formats called containers (boxes).
Just as you might pack cookies, a letter, and a stuffed toy together in one pretty gift box, an MP4 box holds video data compressed with a codec, audio data, and subtitles side by side. Even when two MP4 files look the same from the outside, the way the video inside was compressed (the codec) can be completely different.
You may have had a video where the sound plays fine but the picture is stuck on black. That happens when your device can open the video box (MP4) but doesn't have the right decoder (codec) to unpack the compressed video inside, so it can't play the picture.
Common misconceptions
A file extension like MP4 or AVI is the codec.
MP4 is a 'box' (a container) that holds video, sound, and subtitles together. A codec is the specific technique that compresses and unpacks what's inside that box.
Where you meet it
When you stream a 4K high-quality video on YouTube without stuttering, a modern codec is at work.
When you shoot a video with your phone camera, it is compressed on the spot with a codec like H.264 or HEVC so the file doesn't get too big.
In one sentence
A codec is technology that shrinks video and sound data so it can be sent, then unpacks it again, and it's a key tool for keeping the digital world light and fast.