Alphas are the progenitors (or sources) of a certain line of supernatural beings, aka "the original monster(s)." Eve, the Mother of All, is the creator of each line of monster.
According to Crowley and Dean Winchester in 7.22 There Will Be Blood all of the Alphas except the Alpha Vampire were killed by Castiel with the exception of the Alpha Shapeshifter which was killed by Crowley. With the death of the Alpha Vampire in 12.14 The Raid, the Alphas are officially extinct. In 15.09 The Trap, it's mentioned by a Leviathan that Eve is seeking revenge for the deaths of the Alphas.
Alphas are more resilient to pain than the average monster. They appear to carry little to no weaknesses to weapons that can normally kill their "children." According to lore, all Alphas have a connection with their entire race, which allows the Alpha to locate them. For the Alpha Vampire, this connection also allowed communication through shared visions or telepathy. Existing for thousands of years, they appeared to have vast knowledge on everything supernatural or otherwise, including information on Purgatory.
Alphas have very few, if any, weaknesses. A blade made from the rare metal iridium was shown to cause the Alpha Shapeshifter pain, while silver was shown to not affect it. The Alpha Vampire was also shown to be weakened by dead man's blood, however he required multiple bags being pumped into his body to keep him somewhat subdued; unhooked from the dead man's blood, it would take him less than an hour to return to full strength.
The Colt was shown to be capable of killing Alphas when it was used by Sam Winchester to shoot the Alpha Vampire through the head, killing him.
List of Alphas
First appearing in 6.02 Two and a Half Men, this shapeshifter has gathered a following of its offspring and is apparently generating more shapeshifter children by posing as human husbands and impregnating their wives. When Sam and Dean rescue one of its children, the Alpha Shapeshifter storms the Campbell family compound and kills Mark Campbell, overcoming several gunshot wounds and tranquilizer darts, as well as a silver knife to the heart. It then reclaims the baby from Dean and leaves. This shapeshifter is able to transform in just a few seconds without shedding its skin.
At some point, this Alpha is captured and taken to Crowley's monster prison. During its interrogation, it takes the form of Crowley's meatsuit. Crowley explains that he tinkered with various substances on several lesser shapeshifters to find something that can actually hurt the Alpha: iridium. He stabs the Alpha's shoulder with an iridium knife and threatens to skin its children in front of it. The Alpha continues to insist that it does not know the location of Purgatory, and seems unconcerned with the fate of its children. Saying that the shapeshifters can make more children, he encourages Crowley to kill them. Frustrated, Crowley beheads the Alpha with an iridium machete, killing it.
The Alpha Vampire is first glimpsed in 6.05 Live Free or Twihard, briefly flashing in the minds of its "children" as it commands them to turn more humans, recruit more vampires for its army. Nothing was known about this particular monster, aside from the fact its "messages" can be heard telepathically by all vampires and render them unconscious.
In 6.07 Family Matters, Samuel and his people capture the Alpha Vampire and attempt to interrogate him, but he is extremely resistant to pain and does not give Samuel any information. After Samuel leaves the room where he is being held, he is approached by Dean and Sam and he reveals to them that, though he is the first vampire, he has a mother. He also tells them that Samuel is torturing him for information about the location of Purgatory, which is where the souls of supernatural creatures go. Shortly after their conversation, he escapes his cage and almost escapes completely, but demons appear and take him away on Crowley's orders. It is revealed that Samuel has been working for Crowley, who continues to question the Alpha Vampire regarding the location of Purgatory. He was held in Crowley's monster prison, but escapes before Castiel kills every monster inside, becoming the only living Alpha left.
A year later, the Alpha Vampire makes a deal with the Leviathans to share humanity, unaware that they betrayed him. He begins panicking when his vampires start dying after eating poisoned humans. He is alerted by Emily to the fact that Sam and Dean are coming for him and he captures them. They explain that they need his blood for a Leviathan-killing weapon and that the Leviathans have betrayed him. The Alpha Vampire refuses to believe them until Edgar reveals this to be true and tries to kill him. Sam and Dean decapitate Edgar, and the Alpha Vampire gives them his blood out of gratitude, though he promises to "see [them] next season" which they look forward to.
The Alpha Vampire senses the disturbance of the loss of vampires in the Midwest Region of the United States and sends the remaining nest in that region to the temporary base for the British Men of Letters to send them a message to call off their incursion. While they are assembling their weapons, Mick Davies gives Sam the Colt to kill the Alpha, but he tells Sam that they have none of the Colt's special bullets. Sam shows Mick how to make them from the spell that Bobby used to create the bullets after a helpful tip from Ruby. Mick is able to make the bullets, but the Alpha Vampire comes in to kill him and his associates. With a clever distraction from Mick and Mary, Sam is able to load one of the bullets into the Colt and shoot the Alpha Vampire in the head, killing him.
The Alpha Werewolf is referenced in 6.08 All Dogs Go to Heaven. Crowley speculates that a series of daylight attacks, in which the victim's heart is removed, are werewolf activity and could lead Sam and Dean to the Alpha Werewolf. However, the attacks are actually being committed by a skinwalker. When Sam and Dean were on a hunt in 8.04 Bitten involving pureblood werewolves, Dean wondered if "pureblood" meant the Alpha of the werewolf race.
The Alpha Skinwalker is alluded to in 6.08 All Dogs Go to Heaven. It appears there is plan for a skinwalker pack leader in a town to oversee the recruitment of other skinwalkers in the town, which live with families as the pet dog. On a signal from their leader, the skinwalkers will "turn" the families they have been planted with, thus exponentially increasing the recruitment of skinwalkers.
The Alpha Djinn is referenced in 6.10 Caged Heat. Crowley walks to the cell of the djinn that Samuel captured prior, Brigitta, and asks her if she's ready to tell him all about her daddy.
Alpha Khan Worm
The Khan worm was a monster freshly created by Eve after she was released from Purgatory, and is therefore the potential progenitor of its species. Although it looks like a cross between a simple slug and a centipede, it is a self-aware, intelligent parasite that can enter a person's ear and take control over their actions as well as access their memories. After it leaves its host's body, a black, gooey substance is deposited in the person's ear.