A Minster is a church that is used by missionaries. It is (usually) the first church built in an area they are attempting to spread the word of God to, or the church that sets up missionary projects in other areas. You could call it the missionary's 'base of operations' in a way. York Minster has served in both those roles.
When the (original) Minster in York was built, Christianity hadn't spread through Yorkshire and the people were all Pagans, thus the need for missionaries to spread the faith.
A Minster can be a Cathedral (York being a prime example) however this is not a requirement. The original York Minster was a small wooden church (not much more than a large hut) which was replaced by other churches and finally the current Minster, all on the same site, and serving the same role.
The York Minster is no longer used for missions in the UK, but does support overseas missions.
Source: Taking the guided tour around the York Minster.