Natural bristles are made of animal hair, while synthetic bristles are often made out of nylon, polyester, or a combination of both. There are also brushes with a blend of natural hair and synthetic filament (like that of the Silver Brush Black Velvet series). The advantage of natural bristles is that they can hold a lot of water, which is perfect when you're into watercolors. The synthetic, however, is flexible and springs back, although it cannot hold water as much as the natural can.
Usually, like paper and paints, choice of brush is by user's preference. Some prefer the softness of the natural hair, while others love the flexibility and spring to their synthetic brushes. So if you are looking into buying, I would suggest that you try at least one of each, and see what is better to your liking. One thing I also noticed is that light handed people tend to prefer the softer brushes, while heavy-handed painters tend to prefer the synthetic brushes.
So, I'll be doing a review of my current brushes. From L-R:
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Arts and Crafts Shops in the Philippines
Note: Will update the photo once I got hold of all my brushes.
*Prices are subject to change without notice