You can do Yoga with no props and many of the students in my Edinburgh University class don't have a mat.  However props do help, and I've listed them, roughly most important first.

• A Yoga mat helps you grip and provides padding.  If you are tall, watch out for short mats.  If you the ground is too hard, get a thicker one (6mm) but avoid spongy aerobics mats.
  • I prefer the Yoga Mad studio mats, available in 3 thicknesses.  It's often a little cheaper on Amazon or sometimes I have some I can sell.  The Yoga Mad Warrior mat is a cheaper alternative, but not quite so grippy or hard-wearing. 
  • I am pleased with my rubber mat which is grippy and made from natural materials.  It does smell a bit of rubber especially when new.
  • My favourite mat is a generic black mat, which is thick, wide, long and grippy.  It was reasonably priced, but unfortunately no longer available from the same seller.  It's heavy and stays at home.
  • There are various premium mats that can cost a fortune.  They each have their dedicated fans and you can ask people.
• A blanket can be folded to provide padding of any desired height and shape, although it is less firm than a block.  You can also cover yourself in relaxation and meditation poses.
• Blocks and bricks provide support when the ground seems too far away.  You can also put them between arms or legs to make sure you are hugging in to the midline.
• Straps help when you can't quite reach to bind, or to contain the shape in a pose where you can't find the activation strongly enough unaided.