Hi, I am quite unknown with Arduino, but I need it for a project, but I don't know if my wishes are realistic.
I want to control electromagnetic coils with Arduino board, but I don't know whether it's possible to change the direction of the current through the coils. So to change it from a positive current to a negative current. If not, does anyone have an idea how I could approach it otherwise? Thanks!
Welcome to the forum
In order to do what you describe you need to use external components controlled by the Arduino
Search for information on H bridges for more detail
What electromagnetic coils?
Just simple copper wire coils
Well, both a watch escapement coil smaller than a gnat's eyelash, and a generator coil on a diesel locomotive are both simple and copper. 
So what have you got, really?
The coils I'll be using are about 9 cm long/high and they have a diameter of roughly 3 cm I think. Do you think I'll be able to control them with Arduino? I was also thinking about a Hall effect sensor and connecting that to Arduino, but I'm not sure if that will work out...
10 turns, or 100,000 turns of wire? Any core or bobbin? How much current?
Hall effect devices are often used with Arduino, but what are you using it for?
The critical thing is the resistance of the coil and thereby, how much current it draws when you connect it to the power supply you propose to use.
From where did you get this coil, what are its actual specifications, and for what do you want to use it?
The coils I'm talking about are from Amazon.com
Do you think these coils are usable?
I am creating a miniature model of a train propelled by electromagnetic forces. In the rail I put permanent magnets, and in the train I want to place two coils. I had figured a pattern in which the coils would switch from North, off, South, off and that repeated. With North I mean that the current goes through the coils in such a way that around the coil a magnetic field will be formed with its North side facing the magnets in the rail. With the correct measurements this will create a system in which the train would propel, but I'll spare you the details. The idea is very simple, but the question is whether I can cause the current through the coils to switch in direction, so that the magnetic field around the coils will change. What do you think?
Err, no! 
Not really!
That item is simply a spool of wire, not in any way designed as a solenoid.
You would need a much more carefully designed electromagnet - with a ferrous core - to implement a "mag-lev" style of propulsion.
We - or you - would need to search around for available ready-made electromagnets. (Making your own is not for the faint-hearted! 
)
If you look at the second picture in the ad, you will note the wire is ACTUALLY BRASS wire with NO insulation. So as soon as you try to use it you will have a direct electrical short circuit.
Not at all sure what this stuff is. "Brass" may be just a description of the lacquer colour, alternative to "Gold".
Actual brass wire is uncommon, and stiff. 
Thanks for your responses, I get what you're saying, but it's just that I've seen multiple examples online that also make use of self-created coils. For example this one: 3D Printed Electromagnetic Train ( Part 1 Prototype ) - YouTube
Do you think that they made a mistake too, or that perhaps such an easy coil might actually work?
It is relatively easy to make coils of your own design, size, number of windings, gauge of wire etc but you must use the correct wire
The Amazon link that you posted does not show coils, rather it is for wire that happens to be coiled on a bobbin. It is not even clear whether the wire is insulated
If you read the reviews everyone seems to use it for arts and crafts.
not one mentions using it for electronics.
The wire you want is called Magnet wire.
And to reverse the current just use an H-bridge.
Once you have answered the first question in #8.
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