An organism from the animal kingdom has lateralization if it has external asymmetric body tendencies. It can happen either from having an asymmetric body or from favoring a specific task over its mirror image. Strictly speaking, there only needs to be one task that an animal favors over its mirror image to have lateralization. Although some people define ambidexterity as the ability to write equally well with the right hand or the left hand, a true ambidextrous person for every task has the same amount of skill for doing that task as its mirror image and is ambidextrous even if they're just as skilled at writing forwards with their right hand as writing backwards with their left hand but less skilled at writing forwards with their left hand.
Evolution of snails
Snails actually have species lateralization, not just individual lateralization. The reason why most species are either almost all dextral or almost all sinistral is the following though genetic drift prevents natural selection letting the fraction of the unusual form from approaching zero: the more of one form there is than the other, the harder it is for the rarer form to find a mate than the more common form, but not impossible because other individuals of the rarer form also exist, but since it is harder, they have a greater tendency to be selected against.
No environment is set up in a perfectly symmetrical way so the preference of hand for a task is not always the dominant hand. Babies first get born with both hands pretty much equally good and because of never having a perfectly symmetrical invironment for a task, use both hands pretty much equally often. After that, the difference in skill between hands determines how much more often one hand is used than the other, which in turn determines the rate of chance of difference in skill between hands the the difference in skill multiplies exponentially in a runaway effect until one hand is entirely or almost entirely used for complex tasks. The runaway effect is actually competing with another weaker effect which has a tendency for the difference in skill between hands to divide exponentially with time and that tendency comes from sometimes being in a situation set up in a way to favor one hand over the other so that the way to do it with one hand is not a mirror image of the way to do it with the other hand, for example, reaching something to your left on a high shelf, and seeing a cutting board with celery and a knife on it that was laid with its handle further left than its blade.
Evolution of handedness
Handedness evolved in some species because it makes the dominant hand gain skill faster than it would have if handedness hadn't evolved. By only using one hand for certain tasks, that hand gets practice twice as often and so gains skill faster, which in some situations is an evolutionary advantage.
Evolution against handedness
Evolution of ambisinistry
Wolves evolved ambisinistry because genetic drift favors ambisinistry over handedness.
Evolution of ambidexterity
Ambidexterity in an individual comes from the tendency for the difference in skill to divide exponentially exceeding the tendency to multiply exponentially. In some species, ambidexterity is more favorable than handedness because there's a greater advantage in improving the skill of the less skilled hand than in making the more skilled hand even more skilled. The advantage of improving the skill of the less skilled hand comes from the fact that the individual sometimes faces a nonsymmetrical situation where they're able to do a task but unable to do its mirror image, and the situation is half the time set up to favor the right hand and half the time set up to favor the left hand. One advantage of ambidexterity comes from playing snooker. In some cases the tendency to multiply exponentially may have even evolved to be negative, just like in Self-induced left-handedness because each individual in the species plays with symmetrical tools as a child then encounters nonsymmetrical tasks that need to be done when they're an adult and because skill has a slight tendency to automatically transfer from one task to another task.
It's not known whether or not elephants have lateralization. It might seem like they can't possibly have lateralization because they only have one trunk. However, maybe an elephant has a tendency to be more likely to curl the trunk one way than to curl it the other way to grab something.
Species lateralization occurs when the difference in numbers of individuals that are right-handed and those that are left-handed exceeds that which can be explained by randomness. That difference varies as the square root of the total population of the species.
Evolution for species lateralization occurs when there's an evolutionary advantage to being the same form as the majority such as snail mating and human tool sharing and evolution against it occurs when there's an advantage to being the opposite form from the majority. One advantage of it is fighting because the less common form has more practice fighting with the more common form than the other way around and so has greater skill of fighting the opposite form. Genetic drift can also drive the evolution against species lateralization.