I’m not sure if I can make this any more clear than a Wikipedia article, but I’ll take a shot at it.
Let me back up a little, first.
Atoms consist of nuclei surrounded by a “cloud” of electrons. The nuclei consist of protons and neutrons and those in turn consist of trios of up and down quarks. Photons convey the electromagnetic force, and gluons convey the force that keeps quarks togther (a sort of residue of this interaction keeps nuclei together through the exchange of pions). This takes care of most of the physics and chemistry that we can see, but nature turns out to be a bit more complicated.
The first big wrinkle in the simple picture I sketched above is that there are three “families” or “generations” which provide two sets of copies to the electrons as well as the up and down quarks. So there are muon and tau particles which are exactly like the electron but about 200X (muon) and 3000X (tau) heavier. (Correspondingly, there are charm, beauty, bottom and top quarks.)
Now the second wrinkle is that the electromagnetic force has (for reasons we don’t know yet) a weird twin sibling that we call the weak nuclear force. It is nothing at all like electromagnetism, but is inseperable from it. The combined electroweak force theory dictates by its mathematical symmetry that alongside the electron, muon and tau there must be neutral fermionic partners: electron neutrinos, muon neutrinos and tau neutrinos. There are three force-carrying particles for the weak component of the electroweak force: the positive and negative W particles and the neutral Z particle.
OK, so you have these neutral particles called neutrinos. They are neutral, so the electromagnetic force does not effect them. As leptons, they don’t “feel” the strong nuclear force either. Only exchanges of W and Z particles will effect them. But these particles are quite heavy, so their spontaneous emission-and-absorption through the vaccum can only take place over short distance and time scales (in contrast to electromagnetism, where photons are massless). At the same time, the coupling constant for the nuclear force is tiny (10^-6 compared to 1/137 for electromagnetism and 1 for the strong force). So the weak force is indeed weak and only acts over short distances. This makes the chances that neutrinos will interact with ordinary matter astronomically small.