The most economical transportation between Stockholm and Arlanda airport is the airport coach service:

One-way standard ticket cost is: SEK 80:--

Advance reservations are not required - simply show up at the bus stop, present your ticket, and board. Tickets are purchased at the Flygbussarna ticket counters located at Arlanda and at the bus station in Stockholm Centrum. Combination tickets are also available that include city bus and Tunnelbana (Underground/Subway) transportation within the downtown Stockholm area (Zone 1) for the entire day of ticket validation. Additional charges may apply during early morning hours. Discounts are also available to qualified persons.
Tickets may be purchased with credit card or Swedish currency.

Rail service between Stockholm Centrum and Arlanda Airport is provided by the Arlanda Express and takes 20 minutes from the airport to the central station. Train service runs every 15 minutes during the day and every 30 minutes in the early morning and late evening hours.

One-way prepaid standard ticket price is SEK 160:--
One-way onboard standard ticket price is SEK 190:--

There are two Arlanda Express stops located below the airport terminals.
Train tickets may be purchased at the Stockholm Centrum Arlanda Express ticket counter, automatic machines at the airport, and onboard the train (an additional SEK 30:-- applies if tickets are purchased onboard), and through the regular rail companies. Discount fares are available. Children accompanied by an adult are free, unaccompanied children travel at a discount rate.
Tickets may be purchased with credit card or Swedish currency from the ticket counters and automatic ticket machines.

Taxi service from Arlanda airport to Stockholm:

Taxi Kurir:
One-way fare (to Stockholm city center) is approximately SEK 345:--
One-way fare (to Morgana Hotel) is approximately SEK 460:--
Stockholm Transfer:
One-way fare (to Stockholm city center) is approximately SEK 350:--
One-way fare (to Morgana Hotel) is approximately SEK 475:--

Taxi service from Stockholm to Arlanda airport:

Taxi Kurir:
One-way fare (from Stockholm city center) is approximately SEK 435:--
One-way fare (from Morgana Hotel) is approximately SEK 540:--
Stockholm Transfer:
One-way fare (from Stockholm city center) is approximately SEK 350:--
One-way fare (from Morgana Hotel) is approximately SEK 475:--

There are a number of taxi companies operating in Stockholm, we have listed the two companies that offer the best fixed price rates (from our experience):

Taxi Kurir
Telephone: +46 (0)8 30 00 00

Stockholm Transfer
Telephone: +46 (0)8 33 20 00

The following two companies are reliable and readily available in Stockholm, but typically charge higher fares:

Taxi Stockholm (Only offers limited fixed fares)
Telephone: +46 (0)8 15 00 00

Taxi 020 (Only offers limited fixed fares)
Telephone: +46 (0)20 93 93 93

When picking up a taxi at the airport taxi stand, you are not required to choose the first taxi in line - you may choose the taxi company that you want to use. Taxi Kurir and Stockholm Transfer will typically provide the lowest fares to and from Arlanda airport (We also recommend these two companies within the Stockholm area). All four companies have large taxi fleets and can provide reasonable response times when called by telephone.

Make sure that the taxi fare is agreed upon before you depart (fixed price fares are available from the larger taxi companies, which are significantly cheaper than typical meter rates - especially during commuter hours). It is also a good idea to try and get a fixed price quote when booking a taxi by phone. Never use an unlicensed taxi (taxi is not properly marked or the driver's taxi license is not displayed in plain sight - legitimate taxis have a yellow license plate with black numbers/letters and have taxi company logos painted on the car).
Fares may be paid with credit card or Swedish currency.