What is the difference between eugenics and evolution by natural selection?
Selection is a mechanism of evolution which favours specific forms of traits over others, this can cause the spread of beneficial mutations through a population.

Natural selection is the spread of beneficial traits/genes through populations as a result of the natural variance in their effect on reproductive output (a function of life history traits like lifespan aging and reproductive rates). A classic example is that of Darwin's finches, where different variants of bill length and shape have evolved in populations based on what food resources (types of seed) are available to them and their ability to utilise those resources.

Artificial selection is selection imposed on a population by humans to deliberately affect specific traits. Examples of this would be milk yield in cattle (we want more efficient production and better quality milk), dog breeds (selection on a whole host of traits like colour, beardedness, height, length) and crops like wheat and barley (selected for higher efficiency and  quality output).

Eugenics, coming from the greek for "well-born" or "good" race, is the application of artificial selection principles to humans. Just as with cattle or crops, there is an aim, it may be to increase height, remove genetic diseases, or promote a certain set of traits like blonde hair and blue eyes. Historically we can associate with a number of radical political ideologies, such as Nazism. Even here in Sweden (people tend to think of it as a very progressive and equality driven country) there was a program of eugenics until very recently, achieved by forcibly sterilising or coercing people in to it, for various reasons (racial purity, health, and antisocial behaviour). Here's an interesting example of a girl designated as being genetically inferior because her family had a history of alcoholism (that page gives a good history of eugenics too). The social and ethical problems of eugenics are obvious.

Under normal conditions a persons reproductive output is largely determined by their own choices and actions whereas under eugenics that choice is taken away and made by someone else and specifically with the aim of "improving" the quality of the population.

So in summary:


natural selection is the result of natural variance in fitness linked to traits
artificial selection leads to a change in traits, in a specifically targeted direction, decided and imposed by another organism (usually humans, I know of no other species which does this)
eugenics is practice of artificial selection on humans enacted by humans with the specific aim of improving quality: often for social, economic, or political reasons and is generally accepted as being against human rights these days.