What happens when one partner leaves the business

My wife just opened her business as a small beauty salon for little girls. She came up with the idea and logo for the business but needed more cash to open the business. So she brought in her mom, brother and sister-in-law. Her percentage is 45%, brother and sister-in-law 45%, and her mom 10%.
Now, her sister-in-law just decided to leave the business and transfer all her shares to her husband (my wife's brother). Is this legal? can she just transfer her percentage to anyone? what can my wife do to protect her percentage and ownership of the business?

Assuming that (a) the business is, indeed, a partnership (your use of "shares" is confusing) and (b) there is no partnership agreement, then each partnership interest is personal property that can transferred, partially or entirely, from the sister-in-law to her husband (or anyone else).

I suspect what concerns you is the "anyone else" aspect. One of the benefits of a properly-drafted agreement among owners, whatever the form of the business, is that transfer restrictions can be included, making it more likely that undesirable co-owners can be excluded.

Disclaimer: This information does not constitute legal advice and does not establish an attorney-client relationship.

That would depend on the nature of the shareholders agreement, and demonstrates one of the reasons it is so important to have one.

Normally, a person can give away any of their belongings (including shares in a business), subject to certain legal constraints (often taxation related). A shareholders agreement can dictate how transferring shares works, and how it is (or is not) permitted.