Advisory shares are a type of stock option set aside specifically for early stage start up advisors to reward them for their help in lieu of cash or a salary. Because advisory shares are a stock option and, thus, merely a right to purchase equity rather than be given it or a direct payment, this can also help to avoid a conflict or an appearance of a conflict of interest.
Many start ups often require the advice of someone in or related to the field of business in which they are entering. This can be anything from contact introductions to service provider recommendations to general business advice. However, whatever the advisor brings is generally considered valuable enough by the entrepreneurs of the newer business to be worth rewarding. In this case, an angel investor would not be given advisory shares. Instead, they would receive normal equity because of the money they invested into the business.
Strictly speaking, this is not quite as lucrative as receiving straight equity as whatever shares still need to be purchased but it still results in capitalization for the business and, ideally, profit for the advisor. Also, and importantly, advisory shares are generally not considered part of the capitalization table but rather as part of the employee benefits budget.