Usually your orthodontist SHOULD present you with the choice of braces that you can wear and explain everything to you. It's what happened in my case at least. I was shown Damon Braces, Incognito (lingual braces) and Invisalign and we discussed all my treatment options, what works better in my case and how much everything costs.
To get back to your question ...
The braces that are not self-ligating use something called ligatures (which are very small and colourful 'elastics' that are placed on the bracket to 'tie' it to the archwire. The self-ligating ones (Damon is a very know brand, but there are others, too), doesn't need this, since the braces themselves have a 'sliding door' that 'locks' them on the archwire. Since they don't need the ligatures, you can clearly see the difference.
As a personal preference (and what my doctor offers), I had Damon braces (no ligatures) and it was a good thing. Sure, 'regular' braces are also great, at the end of the day you need your teeth to get into the right position and a good orthodontist will do it regardless of the braces you're using.