I was first introduced to bias binding in high school. The first item I ever made using it was an apron. We used store bought bias binding, which is cheap and convenient. It's cheap for a reason though. Generally, it is plain and boring. Often it contains polyester, which means when you press it, it shrinks and melts.
You can also make your own bias binding. It is easier than you think. I like to use quilter's cotton. It comes in many great colours and patterns, and it doesn't shrink and twist out of shape when you iron it with a scalding hot iron.
Bias binding is basically just strips, cut at 45 degrees to the selvedge edges. The edges of the strips are then folded to the centre. Lengths of strips can be joined together to make even longer bias binding. Pretty Jane has a really great tutorial to make continuous bias binding.
Once you have made your continuous bias strips, you need to fold them. You can use this technique, which I haven't tried myself, but it looks quite clever. Or you can use bias tape makers. Clover make great bias tape makers. Even though they are about double the price of the other brand that you often see, they are well worth the extra money.
There are many different ways to attach bias binding. Angry Chicken has a really good tutorial on how to attach bias tape to your project. You should watch it, if nothing else, you will be thoroughly entertained. I actually prefer (when I remember) to attach the bias first to the wrong side of the fabric, and then fold it back over to the right side, and then top stitch it down. Sometimes, I will fold all of the bias tape to the front of my project so that there is none on the wrong side at all. This technique actually works really well for an apron. The project shown above (sneak peak, its a gift, I will reveal more later) was sewn with the bias attached first to the front and then topstitched also from the front catching the bias at the back.
If you are looking for a project to use up all the pretty hand-made bias tape you make, Nicole Mallalieu has a free tutorial for a bib. One day, when I have one that is completed and is not covered in someone's dinner, I will post a picture here. They seem to be going straight from the sewing machine to the table these days.