Why do magnets attract of repel objects, if magnetic field only chages the direction of particles? If we have an resting iron ball, and we put near a magnet it would be move and accelerate. I'm understanding that the magnetic field is a relativistic effect of electric field, and if I consider magnetic field as just electric, it would be ok. But what about classic physic? How does it explain it?
 A: First, relativity isn't needed to understand macroscopic magnetic effects such as the attraction of large magnetic objects. That was explained and understood well before Einstein's work; the explanation culminated in Maxwell's equations, but the magneto-static parts were well understood earlier in the 19th century.  Relativity changed none of that (it was mechanics that had problems, not E&M)
It's true that there wasn't an agreed-upon microscopic explanation of why bulk iron would "magnetize", but the effect was well understood in terms of behavior:  both the math and material-specific data were in common use for engineering in the 1880's. 
The misconception comes from a statement that you'll occasionally see in a text books now, usually right after the Lorentz force ($\vec{F} = q \vec{v} \times \vec{B}$) is introduced:  "Because $\vec{F}$ is always perpendicular to $\vec{v}$, the Lorentz force can do no work".  That's true for the simple case of an individual electron moving (slowly) in a completely static magnetic field, but it's misleading for the macroscopic case: it's far from the situation of two bulk magnets attracting each other.
The force between two magnets involves a lot of electrons in different places, not just one. The combination of the Lorentz forces on those results in a net force on the magnet which can do work. The traditional diagram for that is:

To say it another way, if the electrons move only under a Lorentz force, no work is done. But if they are also subject to mechanical forces which move them, then that motion may have a component along the Lorentz force, and work can be done. 
But you don’t need to know electrons to understand this. A 19th century engineer or natural scientist ("physicist" wasn't in common use until very late) would understand attraction in a couple of ways:


*

*There's more energy in the magnetic field(s) of two magnets oriented N-S when they're farther apart than when they're closer together: there's a mechanical force that turns that difference into work as the magnets move

*A magnetic dipole in a non-uniform magnetic field feels a force. Each bit of the iron ball is a separate bit of magnet with its own dipole.  Add those up, in the local field, to get the entire force.
Pre-relativity E&M worked with currents and fields ($\vec{E}$) and ($\vec{B}$) that, within the macroscopic domain, worked quite well. They didn't really know about the electron (1897) or the proton/nucleus(1911), let alone special relativity (1907), but they were able to explain and use magnetism quite well: The transformer dates back to the 1830's, for example.
A: I wouldn't say that the magnetic field is a relativistic effect of the electric field. Rather, what for a relativistic observer is a purely electric field, for another observer can be a mixture of electric and magnetic fields. No observer is privileged, so we can't say that what's observed by one is an "effect" of what's observed by another.
You probably know that a magnetic field exerts a force on a moving charge or a current. The force is perpendicular to field and to current or velocity. Materials that are attracted by magnets have sort of closed current loops. If you think about the action of a magnetic field on such a loop you'll see that the forces make the loop rotate (a torque) until the plane of the loop is perpendicular to the magnetic field. At this point the magnetic force tries to pulls the loop apart, within the plane of the loop.
If the magnetic field is constant in space these pulling forces are all equal in magnitude and opposite in pairs, so the loop doesn't move. But the magnetic field of a magnet isn't constant in space: it's weaker the farther we are from the magnet. The force on the part of the loop closer to the magnet is therefore slightly stronger than the one on the opposite side, and it "wins": there's a net force pulling the loop towards the magnet. (This phenomenon has similarities with tidal forces.) The ball of iron is pulled towards the magnet by the sum of all these net forces exerted on each of the "current loops" within it.
The explanation above is very imprecise, it's just meant to give you a rough picture. You can also check for example the Wikipedia articles on magnets and on magnetic moments, especially the part about the force between magnetic dipoles, you'll find a more thorough explanation and further references that you can explore. Please see Bob Jacobsen's answer with its insightful historical remarks. 
A: 
How does [classical physics] explain [magnetism]?

It can't: Bohr–van Leeuwen theorem.
There is, however, a model that's used in secondary schools in Germany (and maybe everywhere else?):


*

*Magnets contain many tiny "elementary magnets" that are all aligned and point in the same direction.

*Iron also contains elementary magnets with random alignments, cancelling each other out.

*In an external magnetic field, the elementary magnets in iron do align and point their "souths" to the external "north". A (temporary) magnet forms and the usual "opposites attract" law applies.

