Whenever you leave a job you should get a form P45 from your employer. When you start your new job, give the P45 to your new employer. Your new employer will use the details on your P45 to calculate your tax.
If you do not give your new employer a P45, you may end up paying too much tax because emergency tax may be deducted from your pay.
If you do not have a P45, use the Jobs and Pensions service in myAccount to register your new job. (Source: Revenue)