What exactly is an anti-neutrino? According to the the definition of anti-particles, they are particles with same mass but opposite charge. Neutrinos by definition have no charge. So, how can it have an anti-particle?
 A: There are other neutral particles with antiparticles, such as the neutron and the $K^0$ meson.  In those cases we have a microscopic theory that says those particles are made of quarks: for instance, the $K^0$ is made of a down quark and an anti-strange quark, while its antiparticle the $\bar K^0$ is made of a strange quark and an anti-down.
The neutrino is different from these because we have no evidence that it has any composite structure.  While the neutrino doesn't have any electric charge, it does have a quantum number that appears to be conserved in the same way as electric charge: lepton number.  We find in experiments that neutrinos are never created alone.  A neutrino is always produced in conjunction with a positive lepton ($e$, $\mu$, or $\tau$), and an antineutrino is always produced in conjunction with a negative lepton.
There is another key property of neutrinos that's important when thinking about their antiparticles, which is their spin.  Weak decays break mirror symmetry (or "parity symmetry"). If you have a beta-decay source that doesn't have any spin to it at all, and you measure the spins of the decay electrons that come out, you'll find that they are strongly polarized: beta-decay electrons prefer to be "left-handed", or traveling so that their south poles point forwards and their north poles point backwards.  Beta-decay antielectrons, by contrast, prefer to be right-handed.  The neutrinos follow the same rule: neutrinos have left-handed spins, and antineutrinos have right-handed spins.
If a neutrino had exactly zero mass, this polarization would be complete.  However, we now have convincing evidence that at least two flavors of neutrino have finite mass.  This means that it's possible, in theory, for an relativistic observer to "outrun" a left-handed neutrino, in which reference frame its north pole would be pointing along its momentum — that observer would consider it a right-handed neutrino.  Would a right-handed neutrino act like an antineutrino? That would imply that the neutrino is actually its own antiparticle (an idea credited to Majorana).  Would the right-handed neutrino simply refuse to participate in the weak interaction?  That would make them good candidates for dark matter (though I think there is other evidence against this).
It's an open experimental question whether there is really a difference between neutrinos and antineutrinos, apart from their spin, and there are several active searches, e.g. for forbidden double-beta decays.
A: Whilst the neutrino is electrically neutral, electric charge $Q$ can be expressed as a combination of (the 3rd component of) weak isospin $T_3$ and weak hypercharge $Y_W$
$$Q = T_3 + \frac{Y_W}{2}$$
For the (left handed) neutrino, $T_3 = \frac{1}{2}$ and $Y_W = -1$ thus, the electric charge of the neutrino is
$$Q = \frac{1}{2} - \frac{1}{2} = 0$$
For the (right handed) antineutrino, the charges are opposite:  $T_3 = -\frac{1}{2}$ and $Y_W = 1$ thus, the electric charge of the antineutrino is
$$Q = -\frac{1}{2} + \frac{1}{2} = 0$$
