What's the difference between normal beer and trappist? What's the difference between normal beer and trappist? Is there any difference in flavour or brewing style?
Trappist beers are all abbey beers, but not all abbey beers are trappist. It's a bit like champagne, it's a protected product name to designate a certain type of classical brewn beers.Trappist is the enumeration of beers brew within the walls of an abbey inhabitted by the Trappist monks of the Tre Fontane. 

Currently (Okt. 2015) there are 11 recognized trappists, of which 10 are actively producing: 

Six Belgian trappists:


Chimay 
Orval
Westmalle
Westvleteren (St. Sixtus)
Achel (St.  Benedictus of the Achelse kluis)
Rochefort


Netherlands:


La Trappe (Brouwerij de Koningshoeven)
Zundert (Brouwerij Abdij Maria Toevlucht)


Austria: 


Stift Engelszell


Italy:


Tre Fontane Abbey


United States of America:


Spencer


There's also one aspiring Trappist brewery in the US named St. Joseph’s Abbey.

There are some strict rules to obtain "Authentic Trappiste Product":


The beer must be brewed within the walls of a Trappist monastery,
either by the monks themselves or under their supervision.
The brewery must be of secondary importance within the monastery and
it should witness to the business practices proper to a monastic way
of life
The brewery is not intended to be a profit-making venture.  The
income covers the living expenses of the monks and the maintenance
of the buildings and grounds.  Whatever remains is donated to
charity for social work and to help persons in need.


There are some standardized naming conventions, for instance the Double (orginally used by Westmalle) is a brown ale which has an alcohol level of around 7%. 

The Tripple is another name first used by Westmalle, which are often blond and have an alcohol level of about 8% to 10%. 

For future reference you can always visit the Trappiste website here.