This is a basic-enough question that you should probably consult some of the many good resources online first and then come back here if you have further questions that could be better answered by a specialist. The Image Archive on the American Eugenics Movement is a great place to start, though as websites go it is getting a little long in the teeth (it was pretty good looking about 15 years ago!). For more in-depth discussion, I am fond of Diane Paul's Controlling Human Heredity but Dan Kevles' In the Name of Eugenics is also very good.

If I were having my students write a very short "identification" on eugenics, it would look something like this:

Eugenics was field of study and policy created in the late 19th century, coming both out of the scientific study of evolution (artificial selection in particular) and human heredity. The field originated with, and had its name coined by, Charles Darwin's cousin Sir Francis Galton, an early pioneer of the study of heredity and statistics. Meaning "good breeding," eugenics was meant to be artificial selection applied to the human race, whereby societies would manipulate reproduction rates to encourage traits it found desirable (like intelligence) and discourse traits it found undesirable (like criminality). Eugenics policies came in two flavors, "positive" eugenics (increasing the "good" genes) and "negative" eugenics (decreasing the "bad") genes, colloquially remembered as "breeding and weeding." British eugenics policies tended to focus on "positive" approaches, but as the ideas got exported to other nations in the 20th century many adopted "negative" policies as well. The United States, for example, adopted three major policies that were justified at times with eugenic rhetoric: compulsory sterilization of the "unfit," restrictive immigration laws for regions of the world deemed to be the source of "unfit" individuals, and anti-miscegenation laws prohibiting "race mixing." While eugenics was nearly synonymous with the study of human heredity in the early 20th century, by the 1930s it had become considerably marginalized within the scientific community and considered more a form of politics than science. Nazi Germany famously embraced eugenics in the form of "racial hygiene" in the 1930s, pointing explicitly to the United States as inspiration and proof of concept, implementing even more expansive sterilization laws, as well as executing the congenitally disabled as part of a "euthanasia" program. Eugenic sentiments and theory, as well as techniques developed in the euthanasia program, were also implicated in the massacres of the Holocaust. By the 1950s, eugenic thought had fallen thoroughly out of favor within scientific circles for the most part (or supplanted with less coercive approaches, like genetic counseling), and by the 1960s and 1970s it was roundly condemned as a form of scientific racism, sexism, and pseudoscience. Interest in the history of eugenics resurfaced in the wake of the Human Genome Project in the 1990s, in part because of the quasi-eugenic statements of the HGP's director, James Watson.

OK, that's probably three times as long as one of their actual identifications, but you get the drift...