Besides being the naked practice of white supremacy in the segregated South, lynching was meant to serve as a warning. It was a punishment by those self-appointed southerners who wanted to maintain a racist status quo designed to discourage others from challening the way things were, or asserting their own civil rights. The fact that so many lynchings occurred without an arrest or conviction just reinforced the message.
The previous posts were very stellar. I would say that one of the most bizarre purposes of lynching was to facilitate social cohesion. If we accept that lynching was a way to intimidate and create fear amongst African- Americans, then one of the purposes was for the perpetrators to coalesce. Lynching was not an isolating or alienating experience for those who perpetrated it. Groups of Southern Whites moved together in lynching African- Americans, allowed their picture to be taken by those who they lynched, and enjoyed a sense of social superiority as a result. If lynching was a response to the desire for African- Americans to actually seek a level of opportunity and advancement, then the cohesion as a result of lynching might have been a response to this desire. In this light, lynching's purpose was to help assert White superiority in a changing social dynamic in the South.
I believe that one of the primary reasons that lynching was done was to instill a sense of fear into certain groups of people. During this time period, most lynching victims were black and it was generally believed (mostly in the south) that they were "beneath" white people. Lynchings were usually carried out by a mob and were often the result of an unfair trial in which usually black people were found guilty. The lynching usually occurred through hanging although sometimes the victims may have been burned. Many times they were often tortured as well.
It is not really possible to answer this with 100% certainty because there could have been a variety of reasons held by the many people who participated. But the usual answer given is that lynching was done to keep blacks "in their place."
One of the major reasons for lynching (or at least one of the major pretexts) was to punish blacks who got out of line. This was particularly true of black men who were deemed to have acted improperly towards white women (Emmett Till being the most famous instance). Whites thought that this was particularly dangerous to their social system because it implied that the black men were good enough to be interested in white women.
So the idea is that black men were lynched to remind blacks in general that they were not supposed to act like they were as good as whites.
Lynching is the practice of punishing a person, usually by hanging, without due process of law, on the basis of accusations made by a mob. This kind of practice has existed in the many parts of the world for thousands of years. But this term is more originated in the USA, probably because of an American Planter Charles Lynch, who in 1700's, along with his neighbors often took law in their own hands to punish people who plundered their properties. The term was originally applied to physical punishment such as whipping.
Subsequently many other American living in western frontier areas practiced lynching because absence or ineffectiveness of legal authority. However, the some people continued to follow the practice even after establishment of law and in defiance of law., rather than to support the law.
Up to about last decade of 19th century, victims of lynching had been mostly whites. However, since then most of the victims of lynching have been blacks. There have been 4,752 reported lynchings between 1882 and 1968. In these incidents 1,307 victims were whites and 3,445 black.
I do not believe that lynchings in 20th century was practiced with the view of achieving any useful purpose. It was mostly emotional outburst of people evoked by a crowd frenzy. Perhaps only common factor in such actions were motives like, hate, envy and greed.