The front and rear derailleurs are used to change gears, just like your car would shift gears. With a bicycle, there is no automatic transmission, well not that I know of. You'll have to change the gears by yourself :-)
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When I first looked at road bikes, I found that for the same bike there were from $200 to $700+ of difference in price just because they have different quality of derailleurs and other components. I asked the salesperson why such a big difference of price. He explained to me that high quality derailleurs will shift gears more precisely, quickly and smoothly. Then he explained to me some other reasons that, at the time, I took them for a sale speech to get me to buy more expensive bikes.
After trying many different bikes as suggested by my bike shop, I could feel some differences between different quality of derailleurs. But at the end, does it matter for me that the things shift gears "precisely, quickly and smoothly" . I didn't really care. But at the end, since I was a pure neophyte in road bikes, I decided that I should trust the professionals. So instead of going for the lowest quality of derailleurs, I chose the one right in the middle between the lowest and the highest.
For the first few months I rode my bike on flat roads so the derailleurs didn't matter much. It was only when I started to train myself to climb steep mountains that I realized how important it was to have good derailleurs. Where I train, there are some hilly areas with lots of successive uphill, downhill, uphill ... I found myself to shift quickly to low gear while climbing, and when going downhill, I would want to shift quickly to highest gear in order to gather speed for the next uphill, and then shift back again to low gears for the climb ... It was only then that I understood why it matters to have good derailleurs.
