Oil is an essential lubricant in your engine. It lets metal press against metal without damage. For example, it lubricates the pistons as they move up and down in the cylinders. Without oil, the metal-on-metal friction creates so much heat that eventually the surfaces weld themselves together and the engine seizes.

Let's say that your engine has plenty of oil, but you never change it. Two things will definitely happen:

Dirt will accumulate in the oil. The filter will remove the dirt for a while, but eventually the filter will clog and the dirty oil will automatically bypass the filter through a relief valve. Dirty oil is thick and abrasive, so it causes more wear.

Additives in the oil like detergents, dispersants, rust-fighters and friction reducers will wear out, so the oil won't lubricate as well as it should. 

Eventually, as the oil gets dirtier and dirtier, it will stop lubricating and the engine will quickly wear and fail. Don't worry, this isn't going to happen if you forget to change your oil one month and it goes over the recommended change interval by 500 miles. You would have to run the same oil through the engine for a long time before it caused catastrophic failure.