Actually, in a large solar flare particle energies can get up to 1 GeV, but the top energy of some particles is not really the issue. The issue is the flux of these high energy particles. A 10 MeV proton or electron pretty much rips through most spacecraft bodies, thus, their electronics are effectively exposed to particles at these energies.
The often associated coronal mass ejections (CMEs) produced in association with large solar flares carry with them enhanced fluxes of >MeV protons and electrons. These blobs of plasma and magnetic fields compress the Earth's magnetic field, which can induce DC currents in our power grids and expose geosynchronous (or GPS, I forget which orbit at the co-rotating altitude) spacecraft to the high levels of radiation. After the CME has passed, the effects are not over as they often induce a geomagnetic storm, which enhances the radiation belts and thus further exposes co-rotating spacecraft to high energy particles (thus the name "killer electrons" for the outer radiation belts).
I will add more later and include some links, but the point is that our magnetic field does a tremendous amount to prevent our lives from becoming incredibly complicated, as Timaeus eluded to.
Updated Version
Actually, in a large solar flare particle energies can get up to 1 GeV, but the top energy of some particles is not really the issue. The issue is the flux of these high energy particles. A 10 MeV proton or electron pretty much rips through most spacecraft bodies, thus, their electronics are effectively exposed to particles at these energies.
The often associated coronal mass ejections (CMEs) produced in association with large solar flares carry with them enhanced fluxes of >MeV protons and electrons. These blobs of plasma and magnetic fields compress the Earth's magnetic field, which can induce DC currents in our power grids and expose geosynchronous (or GPS, I forget which orbit at the co-rotating altitude) spacecraft to the high levels of radiation. After the CME has passed, the effects are not over as they often induce a geomagnetic storm, which enhances the radiation belts and thus further exposes co-rotating spacecraft to high energy particles (thus the name "killer electrons" for the outer radiation belts).
The Earth's magnetic field also helps protect our atmosphere from ionizing erosion. By that I mean that once an atom is ionized and exposed to the bulk flow of the solar wind, it will experience a conductive electric field ($\mathbf{E} = -\mathbf{V} \times \mathbf{B}$) and react like a pick-up ion. The force on the particle from such an electric field can easily exceed the gravitational force, thus freeing the particle from the atmosphere. Without the Earth's magnetic field, the ionized part of the upper atmosphere, called the ionosphere, would increase due to the addition of the solar wind's ionization effects. Currently, only charged particles with energies >10-100 MeV, neutral neutrons, or high energy photons (e.g., UV, X-rays, and/or $\gamma$-rays) are able to reach our atmosphere and contribute to the overall ionization.
It is doubtful that during a pole flip of the Earth's magnetic field that we would completely lose our atmosphere, considering several pole flips have happened in the past. However, the point is that our magnetic field does a tremendous amount to prevent our lives from becoming incredibly complicated, as Timaeus eluded to.
How strong is Earths magnetic field in space?
At what distance? The Earth's magnetic field is roughly modeled as a tilted dipole (i.e., the magnetization axis is tilted with respect to the spin axis of rotation). The magnitude at the magnetic equator is given by the approximation:
$$
\lvert \mathbf{B} \rvert \left( r \right) \approx B_{o} \left( \frac{R_{E}}{r} \right)^{3} \tag{1}
$$
where $R_{E}$ is the Earth's radius, $B_{o}$ is roughly 31,200 nT (i.e., the average field magnitude at the Earth's surface near the magnetic equator), and $r$ is the distance from the center of Earth.
As you can see, by the time you reach ~4 $R_{E}$ the magnetic field has dropped to ~490 nT.
would it be strong enough to attract any sort of magnets back to the Earth?
I doubt one could actually create a scenario to test this as all things in space are orbiting at several km/s. Some spacecraft do use the Earth's magnetic field for orientation/attitude control, but generally these just cause small rotations not radial forces or effective drag forces.
or repel it away if it's facing the similar magnetic pole?
I highly doubt this.
Does that mean a neodymium magnet with the strength of 1.4 tesla and hypothetically the same size of the Earth can protect itself far much better from Solar Wind?
If we had a stronger field and the source region was as large as the Earth, then yes. If you used something the size of a small refrigerator magnet then no. You could see this by changing (in Equation 1 above) $R_{E} \rightarrow r_{o} \sim 1 cm$ and $B_{o} \rightarrow b_{o} \sim 1 T$. The size of the source shrunk by ~8 orders of magnitude while the strength of the field only increased by ~5-6.
The Earth protects us from the solar wind because those particles are charged, thus they respond to a magnetic field. The Earth's magnetosphere also creates a bow shock upstream of the Earth, which provides additional protection since it slows down any inflowing particles and deflects them (due to gradients in the magnetic field).
I wrote some more details on the effects of the solar wind at https://physics.stackexchange.com/a/214509/59023.
Best Answer
It has nothing to do with pressure in the thermodynamic sense nor with virtual particles. There is an intrinsic magnetic field generated somehow in Earth's core (dynamo discussion could fill volumes) and that field interacts with the magnetic field and charged particles of the solar wind. Since the solar wind is supersonic, there is a bow shock generated. This decelerates and deflects the solar wind around the magnetosphere, which stands off from the Earth. Without this, the solar wind's convective electric field (i.e., basically a $\mathbf{E}_{sw}$ = $- \mathbf{V}_{sw} \times \mathbf{B}_{sw}$ field due to the motion of charged particles carrying a magnetic field past the Earth) would drag the ionized upper atmosphere off Earth very quickly.
This is wrong, it does protect Earth's atmosphere from the solar wind, as I stated above. The drift velocity induced by the solar wind's convective electric field on newly ionized particles (called pick up ions) is called the ExB-drift, and it ranges in speed from 10s of km/s to 100s of km/s. The escape speed from Earth at the surface is only ~11.2 km/s. Thus, if the ionized upper atmosphere were suddenly exposed to $\mathbf{E}_{sw}$, the ions and electrons would immediately be accelerated up to 10s to 100s of km/s, easily escaping Earth's gravitational field.