As is normal for tax threads on Ocuk, there is loads of incorrect information here.
First, students do pay tax just like everyone else, you get the personal allowance, which is the tax free amount you can earn. You then pay tax on everything over that amount.
The P38(s) (note the s after it, the P38 is a completely different form) is for students who plan to only work during their holidays and expect their earnings for the whole tax year to be less than the personal allowance, and must be completed beforehand.
The P60 will not state if you are due a refund or not. It will only show yout pay and tax for the year from that employment (and previous employments if you gave that employer a P45 from a previous employer in the same year).
azamc - You will probably be wanting to claim for two tax years. The one ended 5 April 2008, and the one that started on 6 April 2008. For last year, you will need your P60, and any other P45s you received during the tax year, and send them with a covering letter to your tax office asking them to check the amount of tax you paid. Make sure you detail what you did for the whole year, detailing periods when you were not working and wether you were claiming benefits in those periods or not.
For the current year (the one that started on 6 April, you will need to wait until you leave and get your P45. Then fill in a P50, completing the appropriate declarations and send that to your tax office with the P45 to get any appropriate refund.
I would also advise against following mangobreeder's advice of 'demanding your cash asap' as rudeness will only serve to annoy the HMRC employees who are dealing with your repayment claim, and they may feel compelled to make you jump through hoops to get your refund. Politeness is always the best approach.