An NFT is a Non-fungible token.
“Non-fungible” means that it’s unique and can’t be replaced with something else. For example, a bitcoin is fungible — trade one for another bitcoin, and you’ll have exactly the same thing. A fungible token is an asset that can be exchanged on a one-for-one basis. Think of dollars or bitcoins — each one has the exact same value and can be traded freely. A non-fungible object, by contrast, has its own distinct value, like a classic car or piece of art.
At a very high level, most NFTs are part of the Ethereum blockchain. Ethereum is a cryptocurrency, like bitcoin or dogecoin, but its blockchain also supports these NFTs, which store extra information that makes them work differently from, say, an ETH coin. It is worth noting that other blockchains can implement their own versions of NFTs.
NFTs can really be anything digital (such as drawings, music, your brain downloaded and turned into an AI), but a lot of the current excitement is around using the tech to sell digital art. Furthermore, NFTs are designed to give you something that can’t be copied: ownership of the work (though the artist can still retain the copyright and reproduction rights, just like with physical artwork).
To put it in terms of physical art collecting: anyone can buy a Monet print. But only one person can own the original.