How do you write a year with two digits?
This year is 2013, is it OK to omit the first two digits and write "in '13" in English?　Do you always have to write "in 2013"?

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You often hear people talking about 'the summer of '68', or 'the 07/08 financial year', so in certain circumstances it is okay.

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When we write full date such as 24-05-13 (DD-MM-YY), i think then no need to mention whole digit of year but when we talk about only year, we have to mention all digits. 2013 or 2012....

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To be completely correct in your writing, you would add an apostrophe to the short version. Eg: '13. Not just simply 13.

Correct: Britain returned Hong Kong to China in June of '97.
Incorrect: Britain returned Hong Kong to China in June of 97.

You can also find it explained in the Oxford English Dictionary: http://oxforddictionaries.com/words/apostrophe

لا تفوّت فرصة تعلّم لغة جديدة وأنت مرتاح في منزلك. تصفّح مجموعتنا المختارة من مدرّسي اللغات ذوي الخبرة وسجّل في درسك الأول الآن!