Often used to describe blockchains, a system is said to be permissionless when there is no entity that can regulate who can use it and how it can be used.
A system is said to be permissionless when it has no so-called “gatekeepers.” In the case of blockchains, we can say that Bitcoin is permissionless since no entity can forbid anybody else from using it or limiting its use for any purpose.
Put simply, Bitcoin is permissionless because anybody can use it in any way they want as long as the protocol itself allows for it, without anybody being able to stop them.
The permissionless nature of cryptocurrency is often highlighted by Bitcoin proponents as one of its foremost features. Thanks to it being permissionless, Bitcoin and its blockchain can be used by political dissidents or other organizations blacklisted by the banking systems, such as Wikileaks or even criminals. No state can censor transactions on a permissionless blockchain.
Permissioned systems, on the other hand, have some entity which limits who and/or how the system can be used. The traditional financial system is permissioned, since banks and states limit who and for what purposes people can use banks and other financial entities.
Much like the traditional financial system, some blockchains are also permissioned in nature and cannot be used by anyone and for any reason. Usually such blockchains are used by enterprises for a single purpose and can only be used by market participants, as only whitelisted addresses can use the chain.