While many forward thinking individuals care little about "Blood Status" now a days, it is still a very much relevant subject in Magical Society. Blood status is, in a nutshell, "how much wizard you are". Blood Status has been a point of contention for many generations, and many use it to make themselves "superior" to others. It is, most definitely, the most common form of discrimination in the Wizarding world.
There is a popular myth that "Blood purity produces more powerful witches and wizards". This is purely a myth, however and your blood status plays no role in your magical capabilities. The myth is in fact the result of Pureblood families hoarding magical secrets and refusing to divulge them to non-Purebloods.
PURE-BLOOD
Pure-blood families are wizards and witches without known Muggles or Muggle-borns on their family tree. Some have achieved this simply by removing any Muggles or squibs from their family trees and pretending they never existed.
Many pure-bloods consider themselves to be akin to royalty in the wizarding world, or at the very least the elite. They often disdain those of different blood status and Muggles, some even arguing that Muggle-borns should not be admitted into the wizarding world. The Death Eater organisation took this philosophy to an extreme, striving to eliminate Muggle-borns altogether.
To be pure-blood is an uncommon trait and their numbers continue to decline over time. To maintain their "pure-blood pedigree", families such as the House of Black and the House of Gaunt have practiced inbreeding, accounting for the mental instability of certain family members. Those same families tend to disown members who accept Muggles or Muggle-born wizards into their lives. These family members would be deemed "blood traitors". For instance, the Blacks commonly blasted these relatives off the family tree tapestry.
NEW WORLD PUREBLOOD
Magical records were not kept before the formation of MACUSA, and even then they were irregularly maintained (as is common in newly forming governments, especially ones being flooded with immigrants daily.). As such, American Purebloods are often referred to as 'New World Purebloods', which is to say they can only prove their status of all magical ancestors as far back as the creation of MACUSA. Blood purity extremists in Europe will often not accept this 'New World Pureblood' status as being Pureblood at all.
HALF-BLOOD
Wizards with parents or grandparents split between Muggles and wizards are referred to as half-bloods. Due to the dominance of the magic gene, children born to at least one magical parent will usually be magical themselves.
The term 'half-blood' is misleading in that it does not necessarily imply a half-and-half split in wizarding/Muggle ancestry, but rather a witch or wizard with muggle or muggle-born parents or grandparents. By the 1990s, most of the magical population were half-bloods.
The term was used as an insult from members of fanatical pure-blood families. Walburga Black, for instance, was a fervent believer in pure-blood supremacy and her portrait at 12 Grimmauld Place often screamed "filthy half-bloods" and other insults at passers-by somewhat indiscriminately, which points to the mania surrounding blood purity. Bellatrix Lestrange was also known to have insulted Harry Potter and Severus Snape on the basis of their half-blood heritage.
Some half-bloods also express prejudice against those with Muggle ancestry, despite having it themselves, often clinging to what wizarding heritage they do have. Lord Voldemort, the son of a pure-blood witch, Merope Gaunt and a Muggle Tom Riddle Snr, epitomises this. He hated Muggles and Muggle-borns and held his status as the Heir of Slytherin in great esteem.
MUGGLEBORN/NO-MAJ BORN
A child born with magic to two Muggle parents is considered a Muggle-born. In wizarding Britain, they are allowed to attend Hogwarts School of Witchcraft and Wizardry, although some other schools, such as Durmstrang Institute, do not admit Muggle-borns. Salazar Slytherin fell out of favour with the other Founders of Hogwarts because he wanted to limit attendance only to those from all-magic families.
Those with prejudice against Muggles and their families often refer to such wizards as "Mudbloods," a highly offensive term implying that the individual has dirty and inferior blood. Those who discriminate against Muggle-borns believe that they do not deserve magic and should be excluded from the wizarding world, in spite of the fact that Muggle-borns are just as magically talented as those of other blood statuses.
When the Ministry of Magic fell under the indirect control of Lord Voldemort during the Second Wizarding War, it began distributing propaganda against "Mudbloods" under the authorship of Dolores Umbridge. Muggle-borns were also forced to register with the Ministry's Muggle-Born Registration Commission, which accused Muggle-borns of having stolen their wands and sentenced them to Azkaban. This ended once Voldemort was defeated and the Ministry was reformed
SQUIB
Squibs are individuals from wizarding families but have no magic themselves. One example is Argus Filch, who is so embarrassed by this that he hides it from students and studies magic fruitlessly in his spare time. Another example is Arabella Figg, who does not seem to particularly care that she has no skill in magic. Squibs appear to keep in contact with wizarding society, whereas Muggles have little to no contact, as they don't know it even exists.
Squibs are generally looked upon with disdain by most wizards and witches. Even families like the Weasleys, who do not look down on Muggles or Muggle-borns, are rather embarrassed to have a Squib, who works as an accountant, in their family. Neville Longbottom's family was also upset at the prospect that he might be a Squib and repeatedly tried to scare accidental magic out of him when he was a child.
Squibs are able to use magical devices, such as magic detectors, or other items whose magical capabilities are inherent, but not devices such as wands, which require magical abilities in the user.
Muggle-borns are possibly descended from Squibs who married into Muggle families as in historic times went by the affectionate name of 'Magbobs' as the magic 'bobbed up' out of nowhere
DEMI-HUMAN
Demi-humans are half witch/wizard and half magical being offspring, though it can get a bit complicated depending on what their non-wizarding blood is.
Some witches and wizards can be "turned" into Demi-Humans. This means that they can be any blood status prior to turning. These would be Werewolves and Vampires (full, not part). While these individuals are classified as Demi-Humans they often not considered to be "non human" due to the nature of being turned rather then born.
Those that people often think about when they think Demi-Human are the children/grandchildren/etc of Veelas, Vampires, Giants, and Goblins. These individuals often face quite a bit of discrimination for their non-wizard blood and called "Half Breeds".
Though the Demi-Humans that face the most discrimination are Werewolves and Part Goblins. Werewolves are often considered an abomination because of their wild and uncontrollable nature. Part Goblins are considered untrustworthy by regular Witches and wizards and "not Goblin enough" by Goblins.