In a world where Einstein’s relativity is true, space has three dimensions, and there is quantum mechanics, all particles must be either fermions (named after Italian physicist Enrico Fermi) or bosons (named after Indian physicist Satyendra Nath Bose). This statement is a mathematical theorem, not an observation from data. But data over the past 100 years seems to bear it out; every known particle in the Standard Model is either a fermion or a boson.
The fermions are the particles that obey Pauli's exclusion principle, which states that two particles can not have the same quantic numbers. They also have non integer spin. Some common fermions are: the electron, the proton, the neutron and the He 3 nuclei; barions and leptons are included here. They are described by the statistical laws stated by Fermi and Dirac.
The bosons do not obey Pauli's exclusion principle. They have integer spin (or zero). Some common bosons are: the photon, the graviton and the He 4 nuclei; mesons are included here. Bosons are described by statistical laws stated by Bose and Einstein.
An example of a boson is a photon. Two or more bosons (if they are of the same particle type) are allowed to do the same exact thing. For example, a laser is a machine for making large numbers of photons do exactly the same thing, giving a very bright light with a very precise color heading in a very definite direction. All the photons in that beam are in lockstep.
You can’t make a laser out of fermions. An example of a fermion is an electron. Two fermions (of the same particle type) are forbidden from doing the same exact thing. Because an electron is a fermion, two electrons cannot orbit an atom in exactly the same way. This is the underlying reason for the Pauli exclusion principle that we learn in chemistry class, and has enormous consequences for the periodic table of the elements and for chemistry. The electrons in an atom occupy different orbits, in different shells around the atomic nucleus, because they cannot all drop down into the same orbit — they are forbidden from doing so because they are fermions. [More precisely, two electrons can occupy the same orbit as long as they spin around their own axes in opposite directions. What is this spin thing? another article.] If electrons were bosons, chemistry would be unrecognizable!
The known elementary particles of our world include many fermions — the charged leptons, neutrinos and quarks are all fermions — and many bosons — all of the force carriers, and the Higgs particle(s).
Another thing boson fields can do is be substantially non-zero on average. Fermion fields cannot do this. The Higgs field, which is non-zero in our universe and gives mass thereby to the known elementary particles, is a boson field (and its particle is therefore a boson, hence the name Higgs boson that you will hear people use.)
Something else you can do with boson particles is form a Bose-Einstein condensate, a phenomenon predicted by Einstein back in the 1920’s but only produced in a definitive way in the 1990’s, in Nobel-Prize winning experiments described in the link above. What these experiments do in making this condensate is cause large numbers of identical boson atoms to all sit as still as a quantum mechanical object possibly can.
[This is all quantum mechanics, by the way. Einstein didn’t like the implications of quantum mechanics, but you should not have the impression, despite some popular accounts, that he didn’t understand it. In fact his work was crucial in the development of several aspects of quantum theory.]
In principle you could make something similar to a laser out of any boson. This has been done for atoms too. And even more recently, a Bose-Einstein condensate has been made out of photons.
NOTE: Above information has been taken from wikipedia and/or official websites of topics.