If you are employed by a UK company and your tax is calculated/deducted from your payslip via PAYE, then it's unlikely you'll need to do anything at all (I've been working in the UK since I was 13 and I've never had to file taxes). As long as you're on the correct tax code, then your taxes should be correct for the year (I was on the BR code for a while after graduating from uni, but I just gave the tax office a call when I realised it was wrong - they changed the code immediately and I got a refund cheque in the post a couple of weeks later)
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If you are self-employed or meet other certain criteria (i.e. you have overseas income, you earn over £100,000 a year, you get other income from investments or property etc.) you will need to file a tax return (see mapleleafgirl72's link above for the criteria). From what I understand of the process, it is pretty straightforward though, nowhere near as difficult or confusing as US taxes (my dad is self-employed and has to complete a tax return each year).
