Both could be correct depending on the context. The neutral and most common way to say it is to use the simple past - "I celebrated my birthday today." However, if something happened during the period you were celebrating your birthday, you could use the past continuous tense: E.g., "I was celebrating my birthday today when I received news of my sister's accident."
Of course, you can also use the present continuous tense, such as "I'm celebrating my 31st birthday today."
If you are confused as to why it is possible to use the past tense ("celebrated") for *TODAY*, which should be in the present tense, there are at least two reasons why the past tense is justifiable. (1) You are making this statement at the end of the day, when the day is almost over. Then you can certainly say "I celebrated my birthday today." (2) You are referring to a specific event related to your birthday, such as a birthday party, that has already occurred earlier in the day. In this case, you can also say "I celebrated my birthday today."
Hope this was helpful.