There have been four incarnations of the Klan - post-Civil War, early 20th century, Civil Rights period, and Current.
The Klan dates back to the post-Civil War era, when it was founded by Confederate veterans in Pulaski, Tennessee after their side lost the Civil War. Former Confederate General Nathan Bedford Forrest served as the first Grand Wizard of the Klan.
The Ku Klux Klan engaged in such crimes as lynching, intimidation and racial abuse, amongst other things. Despite their infamy for these things, the modern Ku Klux Klan has seen a reduction in the murderous deeds it is known for. (although it still commits crimes like arson and racial hate)
Members of the organisation dress in white robes that cover their entire bodies, with distinctive masks that are said to represent the ghosts of fallen soldiers in the Revolution (in truth this was used to try and scare superstitious black slaves and was as much about concealing identity than anything else - the original Klan didn't wear this uniform), they are also famous for their practice of burning the Christian Cross (despite them believing themselves to be a Christian organisation) - the Ku Klux Klan has been known to set burning Crosses on the lawns of black families in an effort to scare them out of communities.As mentioned above the Ku Klux Klan's past is violent and it has engaged in the practice of lynching - the illegal killing of black men (often via hanging) - this was sometimes used as another means of striking terror in the black community.
Although originally aimed at the freed African-Americans and Republicans later incarnation of the Ku Klux Klan would broaden their hate campaign to include Jews, Catholics, Communists and immigrants.
As well as death by lynching earlier versions of the Klan would burn and shot into the homes of black farmers, sometimes with their occupants still inside and often tried to frighten black farmers out of their own properties - the first Klan was active roughly from 1865 until 1874.
The second Klan (which introduced much of what we traditionally associate with the organization) was set up in 1915 and faded away towards the end of the 1920s after Klan leader D. C. Stephenson savagely raped a young woman, leading to her death. In disgust large numbers of people abandoned the Klan, and law enforcement stepped up efforts to fight the Klan.
The Klan was revived again in the form of the third Klan - which rose up as opposition to the Civil Right Movement and remains (in a splintered fashion) to this day. Despite violent actions by the likes of Byron De La Beckwith, Robert Edward Chambliss, and others, the Klan continued to lose influence and was on the losing end of the civil rights struggle.
But some member can have a change heart and be redeemed
The Ku Klux Klan hoods are very similiar in design to the Capirote, a religious dress worn by members of some Spanish Catholic movements during Holy festivals - dating back to the Inquisition (where it was used during capital punishment) - due to the similiarities people often confuse the two but the Nazarenos (who often wear Capirotes) are not a hate-group and, somewhat ironically, have their basis in Catholicism - while the KKK is anti-Catholic.
The Ku Klux Klan costume is very similar to the costume worn by the Death Eaters only the Death Eater costume is in black
The KKK uniform is meant to symbolize the ghosts of fallen soldier during the Civil War, they often clash with Black Supremacist groups such as New Black Panthers - members of KKK have been murdered by NBP and vice-versa, similar to how early Nazi groups used to war with each other on the streets prior to becoming a national threat.