How to get from the Helsinki-Vantaa airport into Helsinki city centre quickly and without paying too much for it?
Bonus question: how about if the flight arrives in the wee hours (say, at 03:00 am)?
Daytime, there are two very good options:
Late at night & early in the morning (i.e., when 615 is not running), HSL bus 620 is your friend. It runs every half hour mostly, with last departure from airport at 03:55 and from city at 03:10 (Fri & Sat nights 03:40). But first departure is at 05:00 (from city 04:10), so essentially the whole night is covered, all week! Night fare (charged between 02:00-04:30) is 6€. The 620 departs from the exact same platforms as 615, and estimated travel time is 50 minutes.
A taxi is always a convenient option, of course, but it's far from cheap (around 35-40€). There are some fixed rate / shared taxi services too (e.g. Airport Taxi) that are slightly cheaper (from 27€).
Edit: updated for late 2012. Note that slightly different 615 and 620 timetables may apply in the middle of the summer. Check the Journey Planner to make sure, as it always has up-to-date info on the HSL services.