Understanding confluence permission system

Space permission

On space level you define the maximum permission for the whole space. So if you want to permit e.g. single person or groups on (few) sub-pages (viewing or editing) you have to enter these persons or groups on space level. This can be done as a space admin via the area tools at the bottom left (menu "permissions") in the side bar.

Page restrictions

To limit the editing or viewing of individual pages or page trees, you can set "restrictions". This can be done via the lock symbol on a page (on the top bar). As soon as editor or viewers are entered in the menu, the lock turns red. These permissions are inherited recursively downwards, but can be further restricted below if desired. 

How can I open my personal space for others?

By default, the personal space is only accessible to your own user. If you want to make pages accessible for others as described above, they must first be authorized in the space. As soon as this is done, the person has access to the entire pages in the space. If you want only some pages "public" available it is a good idea to create several page trees: e.g. a page tree "private" under which everything you want to keep secret hangs there and a "public" page tree for sharing. The private page tree has then to be restricted to yourself (edit and view) - this makes it invisible to others.  In this way you can also created several page trees and make them accessible to different groups of people.