You will need the Full Write Connections permission to perform this action.
Navigate to the connection record either from the main connection listing or from the person's profile from within the Involvement > Connections tab.
Then select "Edit connection" from the "Actions" menu in the top right.
A pop up will appear where you can edit the details of the connection. See this article for more information on each field.
You will be unable to edit the connection type once the record has been created. If you need to change the connection type, delete the old connection and recreate it using the new type.