Smart Contract Question

Would someone be able to clarify the smart contract specifications for NFT's/Token. I know that they are both created using smart contracts, but my confusion is what is the actual smart contract. For example, when a new NFT is minted, is that actual NFT its own smart contract, and that contract gets traded? Or do NFTs get created from that specific NFT contract ex: Some general Cryptopunk Contract. The same goes with ERC-20 tokens, is each individual token its own smart contract that is managed by a general token contract?