There is no train from HEL to city center, but there are two buses. The city bus 615 is cheaper than the Finnair bus ~ 4 vs. 6.50E and takes about 40 vs. 30 minutes each way. The bus terminates at the main railway station. From there you can walk (or ride tram) through the shopping district and to the harbor, up Mannheim, etc.
www.helsinki-vantaa.fi/ to and from for up-to-date information.
Mid-December can be near freezing or below, with or without snow or rain so walking around may be fun or not.

You will find several hotel recommendations in previous posts. Most recently I stayed at the Scandic Marski, which is a business hotel just a few blocks from the main train station and it was fine.

Can I get to Suomenlinna Fortress in December? If so how long would this trip take?

Yes--
See the website

During winter the Suomenlinna Visitor Centre, Suomenlinna Museum and a handful of restaurants and cafés are open. Guided tours are available in English at weekends. The main route and information boards are lit, but please note that darkness may fall as early as three o’clock in the afternoon. The streets are gritted as soon as possible. However, cobblestones may still be slippery so shoes with good grip are a must. There is currently snow on the ground.

Time--The ferry leaves from the harbor probably once an hour December 30, but you can check the online mass transit schedule for sure. 20 minute ferry ride.

Sorry, even though I previewed my post, I forgot to past the url.
suomenlinna.fi/visitors_guide

Thank you. I also wrote to the visitor center and they also confirmed yesterday.

I will be encountering this same situation this August 2013. I am with a family of 4. We will have 2 long layovers, the first is for 9 hours (morning to late afternoon) and the second is for 15 hours (late evening to afternoon).

I have several questions. This is my first time in this forum so I haven't seen the previous posts.

Which part of the city should I visit and do I have enough time? Should I be concerned since we don't speak the local language? Are there lockers in the airport that will fit some carry-on luggages?

And how was your experience there P67742?

Ramedor88,
If you search, there are a few Helsinki posts where we have tried to help people with this same question.

As you see above, there is no train from HEL to city center, but there are two buses. With 4 people, a taxi may be better. About 30 minutes to city center.

On your short layover you might want to take a taxi to the Church of the Rock, see Finlandia Hall, and walk through the park along Mannerheim, get lunch.

On your overnight, you plan to go to city center and walk from the harbor to the cathedral, orthodox church, along the Esplanade Park, ride the tram. Visit any museum that suits your interest. Or early boat to the fortress. You will still have long daylight in August.

English will be fine in Helsinki.

I don't remember about luggage storage at HEL. There is a day hotel in the airport, so they may handle that. Check the airport website.

Thanks Kay2, I truly appreciate your recommendations.

The day hotel in the airport is too expensive to use as storage. It's also expensive to use them for an overnight stay since I will need two rooms. I will be better off booking a hotel near the city center. Should there be anything to be concerned about with regards to taking a taxi at that late going to the city center? And would it be easy to get a taxi from the city going back to the airport?

A journey by taxi from the airport to the centre of Helsinki takes about 30 minutes and costs around 45 to 50 euros.

Taxis are available 24 hours a day and can easily be booked by your hotel the night before to ensure you get back to the airport on time.

www.helsinki-vantaa.fi/more-information/contacts/contacts_services?group=080

I was able to book a hotel near the airport which includes free shuttle and free breakfast. That solves the problem of transportation. And it's walking distance to the Jumbo shopping mall which will make my wife satisfied and the kids whining.