What are the ways in which a corporation may restrict the transfer of shares?

It's common to think that all shares of any corporation can be treated like any other property once purchased (e.g., sell them, transfer them, etc.). Though this is the default, it isn't always the case. It depends on the rules established by the corporation that issues its shares.

Some corporations choose to strategically restrict their shareholders' ability to transfer the shares they purchased from the corporation to another person. Why? That's a complicated answer with a lot of deep, technical answers. The most simple, basic example would be that some corporations (like small, closely held private corporations) prefer to stay amongst friends than risk letting in strangers. So long as it is reasonable (balancing the importance of being able to transfer shares against the reasoning for why the corporation doesn't want to allow it), a court will allow it. For example, a bunch of family members that form a corporation may want to prevent non-family members or strangers from becoming shareholders of their corporation. How would they accomplish this? There are four primary ways:

(1) The corporation may have a right of first refusal, which means that if a shareholder wants to sell their shares to an outsider, they must first give the corporation an opportunity to buy them for whatever price they were offered by the outsider. If the corporation refuses to buy them, then the shareholder can sell them to the outsider.
(2) A buy-back provision is a contract term that sets a specific price and circumstance which allows the corporation to purchase shares back from a shareholder.
(3) If the corporation is open to allowing outsiders to purchase shares, but wants to be nit-picky about who they let in, they may opt for transfer approval, which requires the shareholder to first get approval from the corporation before selling their shares to an outsider.
(4) Finally, the corporation may opt for a transfer prohibition, which prevents the shareholder from selling their shares to a specific group of people. Note that there must be some limitation on the prohibition, otherwise it would be unreasonably broad. An example of a reasonable transfer prohibition would be a corporation made up of medical doctors that prohibited the transfer to non-doctor outsiders.

The last thing to be aware of is when these types of restrictions are valid. Put simply: they are only valid when the transferee (the person who is attempting to receive or purchase the shares) is aware of the restriction or should be aware of it since it was obviously noted on the share certificate.