Welcome to this mess of a profile. If you don't want to read a mini encyclopedia of Saiyan transformations, feel free to leave this page or browse the images or join the literature club.
There WILL be spoilers if you're not caught up to the Universe Survival arc in Dragon Ball Super just fyi. Also, I won't be covering transformations found only in movies. Now onto the first transformation.
Great Ape

When a Saiyan with their tail sees a full moon, they undergo this transformation. This form multiplies one's base strength by 10. On top of that, the user's consciousness is overtaken and they go on a blind rampage (with Vegeta being an exception). As far as we know, this transformation is exclusive to Saiyans of Universe 7.
Super Saiyan (SSJ/SSJ1)
You might be wondering why there's a 'J' in the abbreviation. The kanji of Super Saiyan is Supa Saiya-jin hence SSJ

This transformation is as synonymous with the series as the Kamehameha. This form is awakened through either a pure Saiyan experiencing intense anger or focusing energy in a certain point on the back. This form multiplies the user's base strength, speed, and power by 50 times (base x50) and makes the user more aggressive in nature. There's several grades of Super Saiyan that I'll briefly discuss. Grade 1 was just explained. Grade 2 buffs the users muscles enhancing power and speed beyond grade 1. Grade 3 further increases strength beyond grade 2 but drastically lowers speed. Grade 4 doesn't enhance power, strength, nor speed beyond grade 1 but staying in the form is more passive and the user is in a calm state of mind so they're more able to focus on strategies in battle.
Super Saiyan 2 (SSJ2)

Awakened by a SSJ through extreme emotional distress or intense training, this form is 2 times more powerful than SSJ (base x100) and gives the user more strength, speed, and power than any of the SSJ1 grades.
Super Saiyan 3 (SSJ3)

This form is achieved by further training and is 4 times more powerful than SSJ2 (base x400). To properly transform, the user needs to control the enormous output of energy. Failure to do so results in the user's destruction. Because of the large amount of ki being used, this form drains a lot of the user's stamina.
Super Saiyan 4

Not canon but this form is achieved by gaining control of the user's Golden Great Ape form. It's not specified how powerful this transformation is, but it's shown to be significantly more powerful than SSJ3.
Legendary Super Saiyan/Super Saiyan Berserk/Berserker Super Saiyan and True Legendary Super Saiyan

A rare and powerful transformation, the user's consciousness is overtaken by the power of the transformation and goes on a blind rampage. Gaining control over the power of Super Saiyan Berserk results in the user being conscious of their actions and thus achieving True Legendary Super Saiyan.
Super Saiyan Anger/Super Saiyan Rage

This is a powered up version of SSJ2 and is awakened by extreme anger. This has been shown to be an extremely powerful transformation as Future Trunks was able to go toe-to-toe with Super Saiyan Blue tier opponents. Because of the large amount of power given by this transformation, it drains a lot of the user's stamina.
Super Saiyan God (SSJG)

This form is achieved through a ritual involving 6 righteous Saiyans where 5 Saiyans funnel their energy into 1 Saiyan or through special divine training. This form is completely different from the regular Super Saiyan transformation line as it utilizes God ki instead of mortal ki and it's extremely powerful. This form was shown to be more powerful than True Legendary Super Saiyan and SSJ3. Despite the power given, this form doesn't drain a lot of the user's stamina. It's still in the air, but this might be due to the properties of God ki. Upon obtaining this transformation, the user can reactivate it at will, but after resting as this form consumes a lot of the God ki required to maintain the form.
Super Saiyan God Super Saiyan/Super Saiyan Blue (SSGSS/SSJB/SSB)

This form is the result of activating God ki along with mortal ki (essentially SSJG+SSJ) and is confirmed to enhance speed, strength, and power beyond SSJG. This is a more advanced transformation than all its predecessors thus the user needs to have the ability to perfectly control his/her ki in order to achieve it. What's unique about this form is that it gives the user more freedom in ki control since they don't have to invest a lot of effort in maintaining the form. This means users can scale the power of their attacks; anywhere from base to the full potential of SSJB. Using SSJB to near or at full potential drains a lot of the user's stamina. This stamina issue can be mitigated by switching between SSJG and SSJB.
Super Saiyan Rose

A variation of SSJB where the Saiyan holds the title of a divine being. This transformation grants the user all the strength, speed, and powers of SSJB.
Super Saiyan Blue Kaio-ken

You can argue that this is more of a technique than a transformation but this is my personal favorite so I had to include it XD. This is only possible because of SSJB's perfect ki control. Combining Kaio-ken with the SSJ transformation line isn't possible because SSJ requires a great degree of ki control to maintain and any attempts to do so results in failure and death. Super Saiyan Blue Kaio-ken stacks the x2-x20 multiplier of Kaio-ken with SSJB which is extremely powerful and has allowed Goku to fight toe-to-toe with godlike opponents like fused Zamasu. Such power does come with risks and drawbacks. Losing control during process of transforming results in the user basically destroying him/herself due to the immense amount of energy being released. After use, the user suffers physical pain to the body due to Kaio-ken, however, a greater degree of mastery can mitigate this drawback.
Super Saiyan Blue Evolution

This is a state achieved by a SSB user who has broken through their limits. It increases the user's speed, agility, and stamina as much as it does the user's strength far beyond SSB as it was shown at full power used by Vegeta, he was able to fight someone with power comparable to a weaker God of Destruction. Additionally, This form can still be taken on even if the user is low on stamina.
Why did I not include...
Kaio-ken? Not Saiyan exclusive and it's more of a technique.
Potential Unleashed? Not Saiyan exclusive.
Ultra Instinct? Not Saiyan exclusive and it's a state of mind.
Thank you and congratulations if you've managed to read through all that. Hope you found it informative. Remember, go beyond your limits.
