While preparing for my lecture on price discrimination, I stumbled upon this Yahoo! question: "Are free desserts for your birthday an example of price discrimination?" The answer appears to be "no." I disagree.
Price discrimination is when sellers charge different prices of the same good to different customers. This is often a way to take advantage of different sensitivities in price. For example, students have more free time than the average person so, with more time to shop around, companies offer them student discounts to entice them to patron their store.
In the case of the birthday dessert, if it's your birthday you generally get to decide where you and your friends will eat. You will probably also want a dessert. Even if you're not paying, you probably have some empathy for your friends's bank accounts. If you can get a good deal, you'll prefer it. And since it's your birthday, you have time to plan which means you are going to be more sensitive to prices (just like a student, who also has time to plan). Thus restaurants give out free desserts, lowering the price of the meal for people who are particularly sensitive to price.
If you need further convincing, note that there is a type restaurant where free desserts is not the case: fast food. These are also places where you are unlikely to insist on going for your birthday meal.