isolation is easy to explain: think shutting the doors and windows, and making them and the walls thick enough so you simply don't hear the outside world any longer!
This is achieved by headphones with the "closed back" design, and by earphones by, for instance, having a foam or silicon plug that fits into your ear.
Noise-cancelling is much more complex and (disclaimer

) I've never tried it, but, the idea is to have a microphone that hears the outside noise and feeds a mirror image of it into your music so that it cancels it. It is said to work well in constant noise, but not for intermittent noises. Good for plane travel, but not so much for shouting children or firecrackers.
As I don't travel by air very much, I've never felt convinced to buy the noise-cancelling type. It adds another level of circuitry and processing for your sound to pass through, and more components to the budget. My principle: let that money be spent on the core business of producing the sound.
There is a third type, of course, which does neither. It is not always appropriate to cut yourself off from the outside world. For instance, it is not safe to do so when driving, or maybe even walking on the roads. using isolating earphones on a bike is just another way of expressing the death wish many bikers seem to have.