When you have a monster as the villain in your story, motivation is unnecessary. The beast can be considered a force of nature, a killing machine. Similarly, a lot of the human monsters, like serial killers, don't need a motivation. They are different from the rest of us, and we can watch their violence with complete belief. As with hurricanes and death-dealing meteorites, we may feel fear, desperation, and anxiety, but we won't feel any empathy.
As I've been digging into villains, many of them do inspire empathy — to the point of becoming tragic heroes, like the original King Kong. Watching one of the first Have Gun - Will Travel episodes, it occurred to me I still worried about the villain, Manfred Holt, long after the story was finished.
Here was a gunslinger who had killed eight men, but he escaped cleverly, cared about his wife and child, was articulate, saved the hero from certain death when he could've gotten away, and held to his own code. He never shot an unarmed man.
The problem was, the slightest offense would lead him to violence, and he gave no consideration for men whose skills with a gun were far below his own. When asked why he didn't just scale his outrage to a fistfight, he said he wasn't very good at fighting that way. Even so, might have escaped the consequences of his actions if he'd been willing to promise he wouldn't hunt down and kill a man who had been a witness against him. He couldn't do that. It would be a lie.
So, even with the kind of villain that is usually reduced to a cartoon, complexity can be worked in, resulting in a memorable character. (Of course, it didn't hurt that Manfred was played by Charles Bronson.)
So, one motivation that can work for a villain is a distorted and inflexible sense of honor.
Another motivation that can create a memorable villain is the need for completion of some sort. This may be tied to a humiliation or a vendetta or an ancient wound. The idea that a group of people must pay for a historical wrong (mistreatment of family members, taking of land, or impoverishment) can drive a villain to what they see as vigilante justice. This can resonate with views of wars between people and provide insights about the human drive for revenge.
Now, the hero might have this kind of motivation, too, but villains usually add a distorting ripple by either making it a grudge that reaches too far into the past or by delivering punishment to an innocent person or meting out punishment that is disproportional.
Trickier is a villain motivated seek to still the voices in his/her head. It needs to be tied to a trauma with which the audience can identify, and often must be presented to them with some immediacy. Again, it must be clear that the victims are innocent or the attacks are out of proportion to the suffering. Getting the balance on the latter right can be very difficult.
The villain may be acting out of loneliness or the need to love. Consider a woman who has been widowed, left without the love of her life. For her to become a stalker, perhaps based on misinterpreting a kindness, could create a distinct and engaging villain.
When the villain is taking on an organization or society so as to be heard, particularly after having made reasonable attempts, people are likely to have empathy, especially if they have had similar experience of exclusion and dismissal. But the direction of the evil acts must be toward representatives who don't deserve the punishment.
Of course, the old standby of any of the seven deadly sins (especially greed) out of control in an otherwise charming person can make for a strong villain. But work is necessary if you want a villain as compelling as Manfred. If a villain goes after a rich person for a small portion of their wealth so that he/she can pay for a child's operation, the balance might shift toward the villain. But explore and test until you find the place near middle point where readers can almost can see the villain's side.
Likability can help gain empathy. Talent and humor can make any character more likable. Even someone as horrible as Hannibal Lector.
Once a good motivation is thought through, it must be presented with human moments. This is often done in good films where just the look on the villain's face tells you that he or she is feeling for the opponent or reconsidering the action or briefly overwhelmed by regret. In novels, too often the writers try and build the case with back story or dialogue alone. Creating a reader experience that is in the moment and based on a gesture that exposes the inner life of the villain is a better way to do the job.