Many organizations and teams try to organize something for a person's birthday as a sign of showing care and interest in their employees. If there is a particular reason that a person doesn't want their birthday to be celebrated (e.g. it might coincide with a sad event), is there an appropriate way to do so without offending or appearing to be unappreciative of this gesture?
Some offices will pick one day of the month and use that as a celebration of all birthdays in that month with breakfast brought in for everyone (bagels, muffins, breakfast pizza, etc. - depends on the season).
This can end up being cheaper than individual celebrations, with the side benefit of being all-inclusive and not putting an awkward spotlight on individuals.
It also makes for a good opportunity for members of different departments in the office to mingle in a more relaxed setting as they collect their monthly free food.
If you'd rather your birthday not be celebrated, either don't let the date be known or, if that information has leaked, simply tell folks "I really wish you wouldn't; it makes me uncomfortable." There's nothing offensive about making such a request. You don't have to explain what the issue is, though if you can do so that may help convince people that you're serious about it.
Care and interest in employees includes respecting their right to privacy.
As keshlam says, the only way is to actually tell people that you don't want it celebrated. If you don't want to tell everybody, you can talk to your boss and maybe one or two more people that you trust, to make sure they can squash any attempt at celebration as soon as they hear of it (which they will but you might not).
You don't need to give a reason other than "I know that everybody means well and I truly appreciate it, but I'd really prefer not to have any type of celebration". This is another instance where "No." is a complete sentence. They don't get to judge whether your reasons are "good enough"; only you can make that judgement and you already have.
If it turns out that your boss and coworkers would prefer to pretend to care (by doing something you have told them you don't want), rather than actually care (by listening to your needs and not pushing something on you against your wishes), it's a clear signal to start looking for a new job. If they don't accept your boundaries on such a personal matter, they won't respect them on other issues either.
Personally, I always book my birthday off as holiday - that way I don't have to be in the office for it.
There's usually someone in the office who takes responsibility for buying a card, ordering a cake etc. Find them and politely explain that you don't want a fuss made. You don't need to go into all the details if it makes you feel uncomfortable.
Depending on where / how you work, see if you can take the day off, or work from home, etc.
Please consider that office birthday celebrations are an opportunity for everyone to take a break, have some refreshments, etc. In some ways, we just use them as an acceptable excuse to party.
Even after considering this, you still feel it is too hard on you, find the organizer and ask them to skip yours. Hopefully, they won't pry into your personal business too much, but you are making an exception to something that everyone else sees as normal. Still, no one should feel forced to participate.