According to Casey Rodarmor's Blog, inscriptions are digital artifacts native to the Bitcoin blockchain. They are created by inscribing sats with content using ord, and can be viewed with the ordinals explorer. They do not require a separate token, a side chain, or changing Bitcoin.
This created quite a stir of late within the Bitcoin community as people started to post NFTs on the blockchain. Many feel this is not in keeping with Bitcoin's mission and is using up valuable block space for something they view as nonsense.
Inscriptions can be images, HTML, or text. They are created by including the content in an inscription transaction. When mined, the inscription is made on the first sat of the first output of the transaction, permanently and inexorably marking it, distinguishing it from its fellows.
Once it is on-chain, the unspent output can be tracked and ownership followed across different transactions as time passes. This allow inscriptions to be traded, bought, or sold.