Understanding the structure of the hadron is of fundamental importance in subatomic physics. Production of heavy quarkonia is arguably one of the most fascinating subjects in strong interaction physics. It offers unique perspectives into the formation of QCD bound states. Heavy quarkonia are among the most studied particles both theoretically and experimentally. They have been, and continue to be, the focus of measurements in all high energy colliders around the world. Because of their distinct multiple mass scales, heavy quarkonia were suggested as a probe of the hot quark-gluon matter produced in heavy-ion collisions; and their production has been one of the main subjects of the experimental heavy-ion programs at the SPS and RHIC. However, since the discovery of J/psi at Brookhaven National Laboratory and SLAC National Accelerator Laboratory over 36 years ago, theorists still have not been able to fully understand the production mechanism of heavy quarkonia, although major progresses have been made in recent years. With this in mind, a two-week program on quarkonium production was organized at BNL on June 6-17, 2011. Many new experimental data from LHC and from RHIC were presented during the program, including results from the LHC heavy ion run. To analyze and correctly interpret these measurements, and in order to quantify properties of the hot matter produced in heavy-ion collisions, it is necessary to improve our theoretical understanding of quarkonium production. Therefore, a wide range of theoretical aspects on the production mechanism in the vacuum as well as in cold nuclear and hot quark-gluon medium were discussed during the program from the controlled calculations in QCD and its effective theories such as NRQCD to various models, and to the first principle lattice calculation. The scientific program was divided into three major scientific parts: basic production mechanism for heavy quarkonium in vacuum or in high energy elementary collisions; the formation of quarkonium in nuclear medium as well as the strong interacting quark-gluon matter produced in heavy ion collisions; and heavy quarkonium properties from the first principle lattice calculations. The heavy quarkonium production at a future Electron-Ion Collider (EIC) was also discussed at the meeting. The highlight of the meeting was the apparent success of the NRQCD approach at next-to-leading order in the description of the quarkonium production in proton-proton, electron-proton and electron positron collisions. Still many questions remain open in lattice calculations of in-medium quarkonium properties and in the area of cold nuclear matter effects.