So, I looked in the website of Vanguard company and saw many different types of ball heads.
I would like to ask what features are must have in a ball head? On what factors does it depend?
There are plenty of factors which must be considered:
I've bought and used two ball heads. I've decided that for me to be happy, it has to have two tension controls on the ball. One sets the basic drag and controls how fast the camera/head combo can move, and the second tightens down so that the camera is locked in position. The first ball head I bought had only a single tension, and when you loosened it so you could move the camera, it could flop all over. What you want is that it be tight enough when "loose" that the camera only moves when you move it, so you can easily do small tweaks of position.
My first ball head did not have any quick release system, you had to unscrew the camera from the ball head. I didn't realize how big a difference it makes until I got my second head with a decent quick release plate.
I did not spring for the Arca-Swiss style release plate system. I'm sure they are cool, but they raise the price of the ballhead far more than I can justify. I got a Manfrotto tripod and ballhead combo, and I love the Manfrotto QR system. Be sure that you can buy extra release plates for your ball head, its important when you have more than one body.
You need to choose a ballhead primarily by the weight you need to support. The load weight that manufacturers report is often difficult to use in making this judgment, though: the manufacturers assessment judges the weight to be perfectly balanced on the head, however depending upon your lens/body combination and how the head is positioned you may have a front-heavy load, for example. A head that claims to hold 5 pounds (far more than a D3100 and 50mm prime) might seem like a good choice, but I would argue it won't be up to the task; I'd say a 10-15lb load is a good choice for typical lenses. Of course, you can always gets something much bigger and heavier than you need -- it'll work well -- but that's added weight for you to carry, a larger size, and no doubt more costly.
Build quality is a huge factor to consider. Even though you've now been able to narrow down which heads are best suited to you, there's still a large quality (and price) range. IMO, what you're really paying for is the enjoyment of the ballhead: a cheap ballhead will move when locking, not lock solidly, be difficult to adjust, not have smooth movement... oh, the agony! A good ballhead will not move when locking, lock solidly and easily, and be silky smooth; a joy to use, and a part of your kit you look forward to working with. And basically, it comes down to this: you get what you pay for.
Most heads are fitted to accept either a 3/8 or 1/4 screw from the tripod. Often they come with a 3/8-1/4 adapter. Make sure the head and tripod are compatible!
If you plan to use the head much, you want to use a quick-release (QR) system. A QR system is comprised of a plate to attach to the body or lens (and leave on all the time), and a clamp on the head to hold the plate. Typically you twist a thumbscrew or twist a lever to lock the plate into the clamp. The gold standard QR system is Arca-Swiss (though there are many manufacturers who make clamps and plates); lesser systems have assorted drawbacks likely to only be recognized after having used them for a while.
I've used Really Right Stuff products for many years and recommend them above anything else. If nothing else, check out their be a gear expert section on ballheads for more in-depth information about various considerations.
The main factors are load bearing, build-quality and attachment.
The bigger the ball head, the bigger the load it can carry, so your current and prospective camera and lenses will inform your decision here. An 'entry-level' ball head should be fine for a small to standard DSLR with anything up to a 70-300mm lens, but this will depend on the quality of the head.
A good quality ball head should exhibit very little 'creep' once its tightened up. Creep is when the weight of the camera makes the ball head move out position very slowly even when its tightened up. Thus, you should be able to get away with a small ball head so long as it's sufficiently well made.
Finally, you should look at the way the head attaches to the camera. If you have a standard DSLR and lenses you will probably be looking at a quick-release plate that attaches to the camera body itself. For bigger cameras, and bigger lenses, you have the option of what's known as an Arca-Swiss type plate, which has a sliding attachment screw allowing the centre of gravity to be adjusted for ultimate stability.