Trains I and P will take you from the airport to Helsinki central railway station. Tickets are available on the station and on board (5.50€), and the train trip will take about 30 minutes. The trains go from 5.30 a.m. until midnight, every 10 minutes almost all day. The entrance to the train station is in the corridor between the terminals T1 and T2 (a few minutes walk from both terminals).
With a bus (415 or 615) the trip to Helsinki takes about 40 minutes, and the price is the same (5.50€).
A journey by taxi from the airport to the centre of Helsinki takes about 30 minutes and costs around 45-50€. The taxi stands are located right outside the arrival areas of the terminals.
More information about the public transportation links to the Helsinki airport and the timetables you can find here.