A zygote is the single cell formed when a sperm fertilizes an egg. It contains all the genetic information needed to develop into a baby and marks the very start of pregnancy.
After ovulation, the egg travels into the fallopian tube. If a sperm reaches it there and enters the egg, fertilization happens. The combined cell is the zygote.
The zygote holds a full set of DNA. Half comes from the egg and half from the sperm. This genetic material guides every later stage of development.
Over the next few days, the zygote moves down the fallopian tube toward the womb. Along the way, it divides again and again to form a ball of cells called a blastocyst. The blastocyst has an inner group of cells, which becomes the embryo, and an outer layer, which helps form the placenta and the membranes that nourish and protect the embryo.
When the blastocyst reaches the womb, it attaches to the thickened lining. This step is called implantation and usually happens about a week after fertilization.
The terms describe stages of the same development. The zygote is the first single cell. The embryo is the stage from implantation through about the first eight weeks after fertilization, when the main organs begin to form. From then until birth, the developing baby is called a fetus.
Sources: MedlinePlus, ACOG.