You 'own' a unique token that is loosely associated with a digital file. This is all stored on a block chain so ownership rights, like BitCoin, can't be manipulated*
The idea is that if you make a piece of digital artwork, such as image, video or piece of code, the origin rights (copyright) are maintained by one person.
A .jpg for example can be copied countless times from the master, but theoretically only one person will only the NFT of the original hence having ownership.
The idea is that it'll be prove those rights (e.g. as evidence for revenue or over disputes of copyright) as the person who made it will have that NFT.
The only issue is that while this all sounds wonderful and could be useful for selling digital rights for anything from tickets etc., it's mostly used illegally.
Moving money digitally around the world is now much, much easier than ever if you know how, but it's not exactly useful for your average buck dealer etc.
*apparently.