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:
HSL bus 615 is easy and affordable. The trip to central railway station costs 5€ and takes some 35 mins. The bus runs every 10-20 minutes between 06:30 and 00:30 on weekdays (Sat 07:00-1:00, Sun 06:05-00:30). (From city to airport, first departure is 05:10 and last 22:45.) At the airport it leaves from T1 (platform 2) stopping at T2 (platform 21) too, and in city centre from platform 3 on Rautatientori, next to the central station. The fare covers transfer to other HSL services – buses, trams, metro, Suomenlinna ferry – within 80 mins. Check 615 timetable or the Journey Planner for more info.
The Finnair City Bus (timetable & route map) takes you a bit faster (average time 30 mins according to the company) from airport to central railway station. The fare is 6.30€, and it runs (mostly) every 20 minutes, between 05:45 and 01:10 daily. (From city to airport between 05:00 and 24:00). This bus leaves from T2 (platform 10) and stops at T1 (platform 1); in city centre, it leaves from Elielinaukio, platform 30, on the other side of the central railway station. The Finnair bus also provides free WiFi (details provided on the information display on the bus), which can be handy for travelers.
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 7€. The 620 departs from the exact same platforms as 615, and estimated travel time is 40-45 minutes. (The 620 also has departures during daytime; the main difference compared to 615 is slightly longer travel time.)
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 that are slightly cheaper (e.g. Airport Taxi, "from 19.50€").
Edit: updated for 2015. 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.
The new Ring Rail Line opens this summer, bringing a rail connection between the airport and central Helsinki. It won’t really be faster than the current buses, but it’s probably easier, more straightforward for the occasional visitor.
As you can see in the schematic map below, you can take either I train or P train no matter which way you're going. To optimise travel time (save 5 minutes), choose I train from Helsinki and P train from airport. Both trains will run every 10 minutes in the daytime.
(NB: at least some bus of the existing bus lines will remain in operation and may be valid options even after the Ring Line opens. I’ll update this with more details later.)