Draft entries allow you to compose an entry in stages. To create a draft entry, create an entry as normal but before submitting the entry, find Publish Status in the Miscellaneous Options on the lower section of the entry creation page. This option defaults to Publish Now.
Change Publish Now to Draft and then save. Alternatively, you may press the Save Draft button at any time.
Your Pending and Draft entries are only viewable by you until they are published.
I'm assuming this issue is not so much with saving the draft but publishing them subsequently? If that's the case, you hover to the right of the title and you'll see a pencil icon which you click (that's the edit). Then when you've finished doing whatever it is you're doing go down to the bottom and change the Publish option to Publish Now. Then press submit (or whatever it says).