Halfbloods are those who are magical, but have some Muggle ancestry. These families form the bulk of the wizarding population of our world, and many of them can probably trace their family tree back to one or more pureblood lines.
Depending on their respective familial situations, Halfbloods may have been raised in pureblooded families and been taught the same prejudices against Muggles and Muggleborns, or they may have been raised to be open-minded and accepting of all blood statuses. They might not hold as high a status as Purebloods in this society, but those with sufficient ambition can still find themselves in positions of power.
Muggleborns are, as the name suggests, the children born of two muggles; in the wizarding world, they are at a distinct disadvantage. Unlike most Purebloods and many Halfbloods, Muggleborns enter the wizarding world oblivious to its cultural nuances: they don’t know about wizarding transportation, laws, animals... The list goes on.
How does this play out? Blood status is the biggest determinant in wizarding society, and it used to be incredibly difficult for Muggleborns to find good jobs in the wizarding world without the help of Halfblood or Pureblood friends. While negative attitudes towards Muggleborns are changing among many witches and wizards, Muggleborns still experience hatred and prejudice, particularly among Purebloods. It remains difficult for them to enter highly-paid professions and is practically impossible for them to rise to positions of power.
Others
Werewolves. Veela. Goblins. Giants. If blood status is the standard by which society judges, then having any non-human attribute or heritage makes just living a normal life in the wizarding world a challenge. As if coping with the difficulties of life (having to deal with a painful monthly transformation, for example) aren't hard enough, letting others know about your heritage is enough to get you kicked out of any establishment and blacklisted from any future jobs.
How does this play out? The 1940s was an era of heightened awareness of differences in Muggle society, and this carried through into the 1950s. Wizarding society echoes a lot of the same mass panic and discriminatory attitudes. Known 'creatures' will be shunned and/or turned away by regular citizens who fear them, and they are likely to become convenient scapegoats for the tensions during the aftermath of World War II. This isn't to say they all have to take things lying down though - who says the Muggle world is the only one with conflict?
Rules Regarding Racism
This site does not and will never encourage racism out of character. That said, we are writing in an era where many such attitudes still abound, especially in the Muggle world. One of the key areas of conflict in World War II (which only ended recently, in 1945) was race, and to roleplay without acknowledging such a rich background would be a shame.
We follow the site guidelines on racism (see here (http://hogwarts-school.net/castle/index.php?page=site_rating) for more), and under no circumstances are racial slurs allowed, whether it's in character in a post or out of character. Your characters are allowed to dislike people for racial reasons and you are welcome to play this out, but we don't want to see the language. There are more creative and subtle ways of going about dealing with such issues and playing out the attitudes of the era, and we encourage you to take this as a challenge.