Closed-chain exercises fix the hand or foot against a surface and move the body, while open-chain exercises fix the body and move the limb freely.
A squat is closed chain: the feet stay on the floor and the body travels. A leg extension is open chain: the hips stay put and the shin swings. Push-ups are closed, bench press is open, pull-ups are closed, lat pulldowns are open.
Closed-chain movements load several joints at once and tend to transfer better to standing, loaded life. Open-chain movements isolate, which is exactly what you want when one muscle needs work the compound lift will not give it.
The old rule that open-chain knee work is unsafe after a knee injury has not held up. Loaded leg extensions in a controlled range are now standard in ACL rehab, and the quadriceps strength they build is hard to get any other way.