The blastocyst phase of a developing embryo takes place at day 5-6 after fertilization. At this phase, the embryo has 70-100 cells and is much more complex than its previous forms.
Traditionally in in vitro fertilization, the embryo is transferred to the uterus on day 2-3 after fertilization. One reason for this is the inability, until recently, to create a culture environment that could house the embryo and facilitate its growth during the next 3 days to reach the blastocyst phase. Medical technology has made great strides in this development and we are seeing much higher percentages of embryos making it to the blastocyst phase in the cultures. 
Blastocysts are more likely to implant than 2-3 day embryos since it is not until day 5 that a naturally conceived embryo makes it to the uterus. At the 2-3 day stage, the embryo is still in the fallopian tubes, and may not be as equipped or ready for implantation as it will be with the added development time. Only about one third of embryos make it to the blastocyst phase, so the ones that do are stronger and better equipped to result in a viable, healthy, full-term pregnancy.
There is a lower chance of multiples when doing blastocyst transfer, because in general fewer blastocysts than embryos are transferred. This is because fewer blastocysts make it to the 5-6 day period, but the ones the do are more viable and have a higher chance of successful implantation. It should be said that there is still a chance that a blastocyst can split and create identical twins, but this is rare and the rates are in keeping with the rates of naturally occurring identical twins.
It is becoming more and more common for blastocyst transfers to be offered as an option for those going through IVF. However, not all clinics offer it. As the technology improves and the necessary equipment to conduct the procedure becomes more available, we should start seeing it more. Talk to your doctor about your options and to get more information about blastocyst transfer.