Public Transportation from Logan Airport is available from both the MBTA's Blue Line Subway service and the Silver Line Bus Rapid Transit service to downtown Boston's South Station (in the Financial District) - Boston's  major tansportation hub for bus, rail, & subway. 
Logan Express Back Bay runs every 20 mins, stopping at each terminal, and serving  Hynes Convention Center and the Copley MBTA Station on Boylston Street.
Logan Express bus departs from all passenger terminals (starting at Terminal A)  on their lower level to: Back Bay, Braintree, Woburn, Framingham, & Peabody.  NOTE, however: As the current Framingham site is under construction until Spring 2015 - Logan Express Framingham is now at a temporary parking facility in Natick (GPS address: 55 Prime Parkway, Natick, MA.)  Scheduled times for Logan departures are shown on Terminal A  where it starts.  Travel time is between 30 & 45 mins. during non-rush hours. 
The Blue Line Subway Line at Airport Station (reached from the Terminals  by free Massport shuttle buses 22, 33 or 55 to 'MBTA Blue Line') connects to the Orange & Green Lines and the Commuter Rail at North Station.  This route is best for Back Bay, Boston College, Boston University, Brookline, Downtown Boston, Fenway Park, Longwood Medical Area, Newton or Northeastern University.
The Silver Line SL1 Bus - Logan Airport / Waterfront/ South Station - departs from outside of each of the passenger terminals.  This route is best direct connection to/from MBTA Red Line at South Station for connection to Cambridge, Downtown Boston, Harvard University, Massachusetts General Hospital and MIT.
MBTA Commuter Boat transportation to either the Quincy commuter boat terminal at Fore River Shipyard, 703 Washington Street, in Quincy, or the Pemberton Point in Hull, is available from the Airport's Logan dock, reached by free Route 66 shuttle bus from all terminals. For more ferry information call 617-222-6999. 

For additional ground transportation information call 617-770-0040.

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