"For educational purpose" or "for educational purposes"?

If you use the singular, you should use an article:

• The university is for an educational purpose.
• The universities are for an educational purpose.

If you don't use an article, you should use the plural:

• The university is for educational purposes.
• The universities are for educational purposes.

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Which of those four variations you use is a matter of context and personal opinion—although the plural noun is probably the more common use.

But a singular count noun requires an article or some kind of pronoun in front of it. (It's possible to interpret purpose here as a mass noun, but it would be somewhat unusual.)