Alpha's fall into four different classes.

Original - A classic definition Alpha. Dominant, Strong, In Charge, Protector
Master - A classic Alpha with a tendency toward sexual sadism. Including bdsm, exibitionism, vouyerism, etc.
Daddy Dom - A classic Alpha with nurturing tendencies. They want to care for their omega more than anything. While all alphas have the potential to be great fathers Daddy Doms are the best.
Supreme Alpha - Classic Alpha ++ Usually has the stereotypical knot, can be very possessive, and often times have Master like tendencies

Omega's fall into four different categories as well.
Original - Sweet, Submissive, Sometimes Shy, Innocent to a point
Pet - Like to crawl on all fours, play the part of a pet, tail, ears, etc.
Little - Have the mindset of a young child. Very Innocent. Playful, Sweet, Caring
Supreme Omega - Classic Omega ++ Has slick, can hold a knot, very possessive, extremely sexual

Sometimes Omega's can have more than one designation, but it very rarely happens. And certain types of Alphas can't have certain types of Omegas.

Original Alphas can have:Original Omegas and Pets. They don't have the patience for a baby, and couldn't fulfill a Supreme Omega

Masters can have: Original, Pets, and Supreme Omegas. They don't have the temperment for a Baby

Daddy Doms can have: Original, Pet, and Babies. They couldn't properly care for a Supreme Omega which would crush them

Supreme Alpha can have: Pet or Supreme Omega. They are too rough for both Babies and Original Omegas.

Alphas:

Sam – Supreme Alpha/Daddy Dom (only one because of the demon blood. No one knows except him Dean and Cas.)
Dean – Original Alpha
Charlie – Mistress
Michael – Master

Omegas:
Gabriel – Supreme Omega/ Baby (rarest combination and only Cas and Michael know about it.)
Castiel – Orginal Omega
Dorothy – Supreme Omega
Lucifer – Pet

Laws:

Laws Restrict the selling or adoption of an Omega that is not a type for the Alpha (ex: A master can't buy or adopt a Baby)

Private Sales of an Omega are the only acceptable sale of an Omega. (There is of course a black market though.)

If an Alpha is unfit the Omega will be instantly removed.

There are training schools and pounds for unruly Omegas. (Basically brainwashing facilities)

Blood tests are given at 16. It tells where you test at. But it only detects a dominant gene due to there being so few dual omegas and no dual alphas. With the exception of Sam for above reasons.