Yes, a comma is required after "May" in this sentence.
The phrase, the 21st of May, is additional information in a grammatical sense for this sentence. For what you want to say, the phrase is not additional information. However, here, we consider grammar and punctuation.
We set off additional information in sentences with commas.
The sentence, without the additional information, is this: "Today is my birthday."
Because the phrase comes after the subject and before the verb, we set it off with commas.
It's also possible to write this:
May 21st is my birthday.
This changes the meaning, as there is a different subject. However, in this case, we do not need commas.
If you were to omit the second comma, you would be saying that today (you have decided that) your birthday is on 21st May (that is to say today you have decided that tomorrow will be your birthday (because where I am, as I write, it is 6pm on 20th May)), whereas tomorrow you may have changed your opinion on the matter and you might decide that your birthday is on 15th April. This would all be very silly, of course, but it just illustrates the effect of punctuation. Omitting the second comma would mean that "the 21st of May" is no longer a parenthetic apposition to "today", and istead "today" sets the scene for what follows: The following is true today. It might not be true tomorrow.
Normally we would just say "Today is my birthday" or "It's my birthday today". There is no need to specify the date, since everyone knows what it is.
If you were writing a story you might want to write "Today, the 21st of May, is my birthday."
'Today' and 'the 21st of May' are the same thing. The writer has decided they are both important pieces of information. In such instances, the phrase is separated by commas. This use of noun phrases is called ' ... in apposition to ...'.
The phrase, 'the 21st of May,' is in apposition to 'Today ...'.
When it occurs at the end of a sentence, there won't be a second comma, just the full stop/period at the end.
"This party is a thank you from me to you all, my dearest friends."