Embryonic stem cell research is the utilization of stem cells from embryos which can then be grown in a laboratory culture and produce specialized cells to treat diseases or used for research purposes. The most common source of such cells is discarded embryos at fertility clinics - the process of IVF typicially produces two or three 'leftover' embryos for each treatment, which are usually dicarded. The use of embryonic stem cells in medicine has met with some controversy, particularly from the pro-life movement that is concerned about the destruction of embryos that they believe constitute human life. This idea, however, is wrong considering that the stem cells to be used for research would have come from embryos that are slated for destruction. This means that if the embryos are not used for research purposes, they will be destroyed. Embryonic Stem Cell research is the leader in saving human lives.
Adult stem cell research is similar except it does not use destroyed embryos, and instead is based on stem cells that can be taken from adults. These cells are less versatile than those harvested from discarded embryos, but they do not raise the associated ethical dilemas  and so far have been easier to work with in achieving results.
The primary reason for wanting to use embryonic stem cells is that they are capable of producing any type of tissue in the body. Thus, theoretically, embryonic stem cell treatments offer far greater potential than adult cells - while both are capable of repairing some damage, embryonic cells could go so far as to replace entire organs. Conversely though, the greater flexability of embryonic cells also makes them more difficult to control. A major obsticle to their use in humans is a tendency to form tumors. Much research is being carried out to better understand the cellular processes which cause this.