General Actions:
A space is a container for pages. It is made of all the pages in that space. In other words it's a set of pages grouped together under the same name. You may have noticed the notation [Space.Page]. This is a link to the page named "Page" in the "Space" space. Thus a page is uniquely identified by both its name and its space name. For XWiki.org for instance, the content is organized within the spaces listed on the right (they are the spaces for this sub-wiki).
This type of organization helps in creating a taxonomy whereby you have the ability to create subdivisions (e.g., server.name/Wiki/XWiki) to better organize information and manage rights more easily on your XWiki.
Note that XWiki currently doesn't support nested spaces (nor nested pages) but pages can have parents. To set the parent for a page click the edit menu and change the Parent field from the "Document Information" panel located on your right.

After you have set the parent for that page the change will also be reflected in the breadcrumb navigation:

Starting with XWiki 2.5 you can also set the parent by clicking the pencil located after the breadcrumbs, in edit mode:


Setting parents to pages allows for easy creation of hierarchies, as deep as you want.
On your wiki's homepage you will see a list of all available spaces:
Spaces are created implicitly and automatically as soon as one page is created with its name as the page space name. For example if you create a link in a page to [NewSpace.Page], the first time you click on the link to create the page the "NewSpace" space will be created and the page will be saved inside it.
Starting with XWiki 2.6, you can create a new space by using the Add menu:

Fill in the Space homepage name and click "Create":

You can also create spaces straight from the homepage by clicking the "Create a new space" link as presented below:
In order to delete a space you will need to delete all pages in that space.