I started reading the book The Women's Book of Yoga and Health and read over a list of items that were outlined as "what you'll need" to practice yoga and I found it a bit long and drawn out. On this list were the following items:
1) yoga mat (preferably sticky)
2) blankets (wool or cotton are great)
3) bolsters (various shapes for multiple uses)
4) straps (reclining, seated or restorative poses)
5) blocks (helps in many ways to create balance)
6) chair (metal folding chair, but kitchen chairs work too)
7) eye pillows & weighted sandbags (eye bags block light to relax and the sandbags apply pressure to specific areas)
First and foremost, I don't own half of these items. I have a mat that I have bought for practice, blankets that I use everyday for lying around, table chair, and I use a belt if I think I need a strap. I've never used a bolster, although I can see how it may be beneficial, and I certainly don't have blocks, eye pillows, or weighted sandbags. These items may enhance a yoga practice for some people but I find that I'm too busy searching for where to put them or if I've placed them right and get distracted from my practice.
A really good sticky (or non-slip) mat is a great tool to have, as well as blankets. Slip-resistant mats help us sweaty palm folks from busting up our faces during down dog, or other positions where hands seem to always slip, and blankets can help enhance a shoulder or headstand tremendously. Extra padding is always nice to have for headstand.
However, I'm not sure I've ever seen someone use blocks, straps, or chairs in an Ashtanga yoga practice. I'm not saying they can't be used, because they most certainly could be, but the practice is geared towards the body finding that happy medium through practice not adjustments made by other objects.
The objects listed above are mostly a person to person desire and by no means a definite answer of what you'll need to practice yoga. Some people will only have a mat to practice with and others will go out and buy a full supply of different items to help boost their practice. No one's cup of tea is the same and if you ask five different people, you may well get five different answers. In the end, you should base your answer on your own decisions. Maybe start with just a mat in the beginning and see if maybe you enjoy a practice in that simple way and if not, then by all means go out and buy the things that will help improve your practice for you.
Yoga is deeply personal and therefore no one can tell you how to formulate your own practice (unless you follow Ashtanga or Bikram Yoga, those are set sequences for the most part). For information on the different types of yoga that is out there for practice, Claudia Altucher has put together a wonderful and informative blog on the different styles. You can read more in-depth about the styles here on her blog.
I hope you find the style you're looking for and create a practice that is both enjoyable and rewarding to practice.