Possibly, your date will mention a party or get-together to which you'll be invited. If you go, treat it as you would the birthday of a friend you very much like: bring a card, buy them drinks, etc. Be sure to mingle with people, and don't expect to spend the whole night by the birthday person's side. You'll probably go home together at the end of the night, but if it doesn't happen, for whatever reason, don't pout. This night is not about your burgeoning relationship.
If your date doesn't have plans, you can offer to take him or her out for a birthday drink, but be wary of this. Sure, it's possible they're just not big birthday people, in which case it's perfectly fine to spend the day together without making a huge production out of the birthday. But beware of people who expect you to make big plans for their birthdays too early on in the relationship -- this screams high maintenance, and besides, don't they have friends to do this? Red flags...
If You've Been Dating Three Months or Fewer, Not Exclusively
Same rules apply as above in terms of party behavior: if you go to a celebration, be prepared to buy drinks and go home together. A card is necessary, a token is nice but not mandatory. Buy breakfast the next morning, make sure he or she isn't too hungover, mingle with friends, etc.
If you're invited to a more intimate celebration, such as dinner at a restaurant with friends, don't feel obligated to attend. Of course, it's lovely if you do—but if you don't feel you're quite yet at the Birthday Dinner With Friends level yet, you can offer instead to take him or her out for a birthday drink on your own (possibly at a later date).
If you don't participate in a group celebration, some sort of acknowledgement about the birthday is still necessary. Next time you see the person, have a card, or offer to foot the dinner or drink bill in honor of their birthday.
If You've Been Dating Six Months Or Fewer, Exclusively
So you're a couple, albeit a relatively new one. Ask early on how they want to celebrate, and plan accordingly. (Maybe they want their birthday ignored all together—but you've got to ask.)
