Yes and no. Let's assume you have a Mac. If you have a Mac and you've logged in to iCloud on both your Mac and your iPhone and you have photostream enabled on both your Mac and your iPhone then yes your photos are backed up. For a while.
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To get them backed up permanently, you need to launch iPhoto and allow it to import them. Once again for about the (third? fourth?) time you must enable photostream in iPhoto having already enabled it on your iPhone and your Mac System Preferences (first 2 screenshots above). I do NOT recommend letting iPhoto push photos to your iPhone from your Mac. As the packrat owner of about half a terabyte of photos, I must always leave this option turned off as shown in the screenshot above "automatic upload" is unchecked.
Photos sit on your Mac somewhere in ~/Library indefinitely but on Photostream they only live for 30 days or 1000 photos so be sure to go on your Mac and launch iPhoto from time to time and allow it to import and make events for the photos you have sitting in Photostream.
To tell you the truth, the new Amazon Fire Phone promises to handle photos better than Apple does. At the prices we are paying for Apple gear, unlimited cloud photo storage is something Apple should have offered iOS users before Amazon did. iCloud storage has gotten cheaper but it's far from a good deal. By comparison: Google gives 15 GB for free as does Skydrive. Meanwhile Apple gives a mere 5GB for free and charges about $24 a year for a total of 25 GB. Also note that photos sitting in shared photostreams do NOT count against your iCloud storage limit (the last time I checked). There is nothing wrong with sharing everything only with yourself simply to back it up.
I don't recommend letting your pictures simply sit in iPhoto Library forever. Eventually the iPhoto Library will grow to multiple gigabytes. While you can always open it using "show package contents" in finder, I prefer to keep iPhoto Library a reasonable size. To do this, I go in several times a year and drag photos out of those auto-created "May 2012 Photostream" events to finder. I then have originals sitting on my OSX filesystem. I then import them back in to iPhoto having turned off "copy photos to iPhoto Library". This gives me a smaller iPhoto Library that mainly has metadata about faces and places and I must manage and back up all the originals myself. Of course once I've imported them to iPhoto, I must resist the temptation to move the files around or iPhoto will not be able to find them.
If you're just getting started, it could be many months or even years before you need to think about this. I'm just bringing it up now so if you go look in ~/Pictures one of these days and notice that iPhoto Library has grown to tens of gigabytes, you know there is a way to shrink it back to a more manageable size.
Hope this helps...