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What does your character want? Can the bad side give him/her that or at least convince him/her it can? In the Games of Thrones television series, character Stannis Baratheon wants to be king. He comes to be under the influence of red priestess Melisandre, who convinces Stannis to partake in more questionable actions to gain more power. Eventually, he sacrifices his daughter Shireen by burning her alive.
What does your character fear? Can the bad side give him/her a way to overcome it or at least convince him/her it can? In the Star Wars prequels, Anakin Skywalker fears his wife Padme dying in childbirth after seeing visions of it in his dreams. The evil Sith Lord Palpatine tells Anakin there's a way to keep Padme alive in order to manipulate him into joining the Dark Side.
Is there a reason your character might disagree with Team Good? Is their approach/goals something the character might feel strongly about? It could be a complex moral issue without a clear cut answer like: freedom vs security/safety, strength vs diplomacy, democracy vs quick action, law/order vs morality, etc. Do the ends justify the means (leads into the next question)? Is Team Good unable to do what "needs to be done"? In Captain America: Civil War, one of the central issues that divides the characters is the question of whether or not they should accept more government involvement and restrictions of their power.
Does your character have flexible morals or is willing to do things "for the greater good" or so that others don't have to? Or does the character feel he/she "has no choice"? In the series Animorphs, character Rachel becomes more violent and other characters specifically call on her to do more heinous tasks such as the killing of another child. She is able to do this and still feel somewhat OK because she is on "Team Good" and is thus "still good".
Has your character been negatively impacted by recent events or negative experiences from his/her past? In the A Song of Ice and Fire universe, Aerys the "Mad King" is rocked by the deaths of some of his children and a brief uprising in which he is held captive for a length of time. This causes his paranoia to increase and he becomes more and more cruel. A perfect storm for madness. His choices are poor, but we can somewhat understand what drove him to this. Revenge for a personal wrong is a powerful motivator and feeds into the next question.
Is there a character he/she has a personal vendetta against? In the manga My Hero Academia, one character is treated poorly by a hero character. This causes the character to become a villain and his main goal is to kill this hero character. I'm oversimplifying a bit, but that's the gist of it.
Does your character have a fatal flaw? Hubris, pride, lying? Or does the character have a virtuous trait that could lead to villainy? In Breaking Bad, Walter White just wants to make money for his family. That's not inherently a bad thing, but it's how far he'll go to achieve his goals that makes him a villain. Each time something goes wrong in response to his bad actions, he has to increase the violence to maintain his control. Maybe the character's bad choices snowball into something much bigger.
Has your character been groomed, lied to, or otherwise manipulated? A negative influence over a character can be very powerful and can overcome the good influences in the character's life. In She-Ra and the Princesses of Power, Adora believes she's on team good until she's outside the Fright Zone and realizes she's clearly with the "bad guys". Prophecy can also come into play here since they can give rise to half-truths and murky uncertainties.
Is there a dramatic misunderstanding? Something like how if only the friar had delivered the message in Romeo and Juliet or if only Guts and Griffith had had a conversation, things might have gone differently.
A lot of these example characters have multiple factors going into their descent into morally grey/villainous behavior. Childhood trauma, someone else saying it's ok (you're doing the right thing, hell maybe said character ALSO things it's the right thing and you've got a cycle of abuse going), positive reinforcement for more negative acts, things blowing up in the characters face (I tried your way, now I'm gonna do this), split-second decisions, and stuff like that.
You could also have the bad guy blackmail him/her or hold someone as a hostage to convince him/her to join, but that's not exactly great for long-term loyalty. Same deal if your character is changing side for his/her own personal safety cuz he/she will just change back depending on whatever is safest.
You could have your character be sworn to another. Not to use another ASOIAF example, but during Robert's Rebellion, some characters only fight for a side because they swore an oath and won't consider breaking it since it goes against their morals/the taboos of society. Maybe the character is super loyal to certain things (faith, family) and the villain fulfills this, so the character is swayed to follow them.
A few other motive to consider alongside the aforementioned questions: ambition/a desire to better ones station (doesn't even have to be for bad reasons, can be for very virtuous reasons!), desperation (if I don't do this, something bad will happen), rivalry, persecution (when I rule, I'll change the laws so that life for peasants is better/vampires and humans can get married/etc), self-sacrifice (this guy has to go down and I don't care who I have to work with to do it), betrayal, or protection (I will do anything to protect my family). Truly, the sky's the limit!