What does it mean to be a blockchain node operator? Are there any blockchains out there that don't have node operators?

A node operator runs software that keeps a full copy of the blockchain and broadcasts transactions across the network. Nodes are needed in order to make blockchains work.

An operator ensures that nodes run with enough resources to keep nodes stable and performant. Nodes need to have enough RAM, disk space, bandwidth, etc. to stay operational and serve the network.

A masternode is a special type of collateralized and incentivized node that offers additional services to the network in exchange for a portion of block rewards and fees. (Examples are anonymous transactions, instant confirmations, file storage, encrypted messaging, etc.) Masternodes must stay operational at all times in order to earn rewards. A masternode therefore is typically deployed on a VPS (Virtual Private Server) with a static IP in a data center.

Since new software versions are often released (with new features, bug fixes, and security enhancements), node operators must maintain current versions of software. Additionally, if a node should fall out of sync with the network (through an inadvertent fork of the blockchain, for example), a node operator might have to resync the node and get it back on the proper chain at the correct block height. A node operator needs to stay up to date with developments (usually through social media channels) to understand the latest status and requirements of the blockchains they support.

Additionally, a node operator might run nodes on behalf of clients, so contract terms/commission rates must be negotiated and implemented.

As for blockchains without node operators, there are permissioned blockchains that don't allow public node operators, but have private node operators nonetheless. And in DPoS (Delegated Proof-of-Stake) networks, there are limits to the number of nodes allowed to append blocks to the blockchain, but still have node operators. In fact, these operators, known as delegates, have even more responsibility because they are chosen by the community to run nodes that secure the blockchain.

Like other stakeholders in the network (ie. miners, stakers, developers, and users), node operators are vital to the operation, security, and health of blockchains.