What does the date of today mean?

It means What is the month and day of today.

How do you ask for todays date?

If you want to know what day of the year it is (answer 2), you can ask What is the date today? or What is todays date?.

Is it OK to say todays?

In most situations, the word “todays” is correct. The apostrophe can indicate possession or omission, but its application will depend on the context in which the word is used.