The J/ψ is a subatomic particle, a flavor-neutral meson consisting of a charm quark and a charm anti-quark. Mesons formed by a bound state of a charm quark and a charm anti-quark are generally known as "charmonium." The J/ψ is the first excited state of charmonium (i.e, the form of the charmonium with the second-smallest rest mass). The J/ψ has a rest mass of 3096.9 MeV/c2, and a mean lifetime of 7.2 × 10−21 s. This lifetime was about a thousand times longer than expected. 
Its discovery was made independently by two research groups, one at the Stanford Linear Accelerator Center, headed by Burton Richter, and one at the Brookhaven National Laboratory, headed by Samuel Ting at MIT. They accidentally discovered they had found the same particle, and both announced their discoveries on November 11, 1974. The importance of this discovery is highlighted by the fact that the subsequent, rapid changes in high-energy physics at the time have become collectively known as the "November Revolution".
Richter and Ting were rewarded for their shared discovery with the 1976 Nobel Prize in Physics.
Background to discovery
The background to the discovery of the J/ψ was both theoretical and experimental. In the sixties, the first quark models of elementary particle physics were proposed, which said that protons, neutrons and all other baryons, and also all mesons, are made from three kinds of fractionally-charged particles, the "quarks", that come in three different types or "flavors", called up, down, and strange. Despite the impressive ability of quark models to bring order to the "elementary particle zoo", their status was considered something like mathematical fiction at the time, a simple artifact of deeper physical reasons.
Starting in 1969, deep inelastic scattering experiments at SLAC revealed surprising experimental evidence for particles inside of protons. Whether these were quarks or something else was not known at first. Many experiments were needed to fully identify the properties of the subprotonic components. To a first approximation, they were indeed the already-described quarks.
On the theoretical front, gauge theories with broken symmetry became the first fully viable contenders for explaining the weak interaction after Gerardus 't Hooft discovered in 1971 how to calculate with them beyond tree level. The first experimental evidence for these electroweak unification theories was the discovery of the weak neutral current in 1973. Gauge theories with quarks became a viable contender for the strong interaction in 1973 when the concept of asymptotic freedom was identified.
However, a naive mixture of electroweak theory and the quark model led to calculations about known decay modes that contradicted observation: in particular, it predicted Z boson-mediated flavor-changing decays of a strange quark into a down quark, which were not observed. A 1970 idea of Sheldon Glashow, John Iliopoulos, and Luciano Maiani, known as the GIM mechanism, showed that the flavor-changing decays would be eliminated if there were a fourth quark, charm, that paired with the strange quark. This work led, by the summer of 1974, to theoretical predictions of what a charm/anticharm meson would be like. These predictions were ignored.
The work of Richter and Ting was done for other reasons, mostly to explore new energy regimes.
Because of the simultaneous discovery, the J/ψ is the only elementary particle to have a two-letter name. Richter named it "SP", after the SPEAR accelerator used at SLAC; however, none of his coworkers liked that name. After consulting with Greek-born Leo Resvanis to see which Greek letters were still available, and rejecting "iota" because its name implies insignificance, Richter chose "psi" - a name which, as Gerson Goldhaber pointed out, contains the original name "SP", but in reverse order.  Coincidentally, later spark chamber pictures often resembled the psi shape. Ting assigned the name "J" to it, which is one letter removed from "K", the name of the already-known strange meson; possibly by coincidence, "J" strongly resembles the ideogram for Ting's Chinese name (丁).
Since the scientific community considered it unjust to give one of the two discoverers priority, most subsequent publications have referred to the particle as the "J/ψ".
The name charmonium is occasionally used for the J/ψ. This is by analogy with positronium, which also consists of a particle and its antiparticle (a positron in the case of positronium).
When referring to excited states of the J/ψ (that is, bound states of a charm quark and a charm antiquark with higher rest mass), the particle is identified as the ψ(mass in MeV). For example, the first excited state (previously called the ψ'), is called the ψ(3686) because its mass is 3686 MeV. The "J" is not used, since Richter's group alone first found excited states.
In a hot QCD medium, when the temperature is raised well beyond the deconfinement temperature, the J/ψ and its excitations are expected to melt. This is one of the predicted signals of the formation of the quark-gluon plasma. Several heavy-ion experiments at CERN's Super Proton Synchrotron may have shown this phenomenon.
* Weak Interactions with Lepton-Hadron Symmetry. The GIM Paper.