The neutralino, usually denoted by χ
0 (sometimes with a
tilde on top), is a
superpartner, one of the extra particles that arise when one has a
supersymmetric theory. Its name is a rather ugly compromise which I will explain a bit later: suffice it to say that despite appearances it has little relation with the
neutrinos. However, like the neutrino, it is an electrically
neutral fermion.
The principal scientific rôles of the neutralino are, to be produced in accelerators, and to provide a candidate for the dark matter making up about 23% of the mass density of the Universe -- specifically cold dark matter in the form of WIMPs. To date, no neutralinos have been found in either situation. This allows us to put limits on the mass of the lightest neutralino and on its interaction cross-section with matter. For what it's worth, the mass limit is currently about 40 times the mass of the proton. Searches continue.
More precisely, supersymmetry requires that for each quantum field representing a particle in the Standard Model (SM) one should add another field with a spin differing by 1/2. So, the SM fermions (spin-1/2) such as the electron get scalar field (spin 0) partners called selectrons, whereas the SM bosons, namely the photon, W boson, Z boson (spin 1) and Higgs (spin 0) get fermion partners (spin 1/2). In order to have some hope of agreeing with observation, the superpartner masses should be different from (and usually greater than) the SM particle masses. When this is done, concentrating on the SM bosons, we get a lot of massive fermions called the photino or Bino, Wino, Zino and Higgsinos. The "ino" suffix comes from analogy with the neutrino, since they're all fermions; the bino and zino are neutral, the winos are charged, and the higgsinos are both.
But also recall that the SM particles get mass through spontaneous symmetry breaking, which gives all the particle masses in terms of the vacuum expectation value of the Higgs. Don't worry if you don't recall this, it just means that there is another source of mass out there which we have to add in. But crucially, the new source of mass mixes up these new fermions. As if it weren't complicated enough already. So in order to get to a set of particles with well-defined masses (mass eigenstates) we need to take mixtures (linear superpositions) of the bino, wino, zino and Higgsinos. Neutral particles can only mix with neutral and charged with charged, since electric charge is conserved.
The result is four neutral massive fermions called neutralinos and two charged massive fermions called charginos (χ+ and χ-). I guess they just ran out of interesting names.
Now the particle that's of most interest in cosmology for that dark matter is the lightest neutralino χ01. This is because the heavier ones decay quickly into the lighter ones, whereas the lightest one can be stable: in many models it's the Lightest Supersymmetric Particle or LSP and is automatically stable. The behaviour of the LSP depends on what mixture it is -- in some cases it will be "mostly bino", sometimes "mostly higgsino" or even "mostly wino". Yes, this is what particle cosmologists say. (Note: by "mostly wino" they mean "mostly zino", since the winos are charged, but for some reason the zino is treated as a honorary wino. Did I mention that the terminology often doesn't make much sense?)
Neutralinos are not heavier versions of neutrinos, or supersymmetric partners of them. (That much should be clear from the fact that they're both fermions.) Neutrinos have nonzero lepton number, which in practice means they are related via the weak nuclear force to charged leptons (electron, muon, tauon); both are fundamental fields in the SM. Neutralinos are related by the weak nuclear force to each other and to charginos and are mixtures of superpartner fields in the MSSM (Minimal Supersymmetric Standard Model) that have zero lepton number. The superpartners are also distinguished from SM fields by R-parity, which means that superpartners can only be produced or destroyed in pairs. This adds up to the result that there's no way that you can turn a neutralino into a neutrino. So they are different types of particle. (Actually, there are some models in which both R-parity and lepton number are explicitly broken symmetries, so the neutrino and neutralino could mix. However, these are regarded as relatively inelegant and unlikely. For example, they don't have a cold dark matter candidate any more.)