How to show content for posts of a specific category only

You can use the in_category function to test if the current post in the loop is in the specific category you want to display full content for.

For example

if ( in_category( 'my_category') ) :
    the_content();
else :
    the_excerpt();
endif;

For more information on the in_category function see the WordPress codex at: http://codex.wordpress.org/Function_Reference/in_category