Sorry that in English, but I don't know enough Polish to explain this.
NI number is needed to record your National Insurance contributions. In time you may be entitled to benefits (council tax, out of work benefits, disability) if you have paid for so many (think 2) years.
NI also counts towards the 49 working years needed for an old age pension (65+). If you earn less than £82 (gross) per week you do not pay.
Income tax is payable by everyone who earns more than the personal allowance. For 2005/6 (year counts from 5th April to 4th April) it is £4,895. That means that if you earn more than £4895 (Gross) in a year, you should pay tax on the rest. It does not make any difference whether you are a part-time student, full time student or have another job.
Also most important - even if you don't earn enough to pay NI or income tax, you must always have a payslip from your employer showing what you have earned and paid out.