Knitting needles are typically split into two categories: straight, and circular.
Straight needles are used for knitting flat portions that you can then sew together. They can be found in single point and double point. The single points are what usually comes to mind when one thinks of knitting (the two sticks). The double points are straight but have a point on either end, and are often used in a triangle for things like sleeves and socks. You can also get cable needles, which are double pointed with a curved body and used for cabling designs on things like jumpers.
Circular needles are used for knitting flat as well, but also for knitting in the round. They can be found in fixed and interchangeable. Fixed circular needles come with a needle attached to either end of a cable that cannot be detached. The interchangeable ones have detachable needles, so you can mix and match cable lengths and needle widths depending on your needs.
There is a variety of different materials available, such as: laminated birchwood needles; stainless steel needles; bamboo needles; aluminium needles; brass needles, etc. Which one to use depends on personal preference.
Most needles are rounded through the shaft, but some have a square grip.