I'm going to keep this to names only in the interest of minimizing spoilers.
#Key
#: Only availiable as a continued romance if initiated in Mass Effect 2
^: Same Sex Romance option
%: Not a Squad Member
$: Does not count towards 'Paramour' achievement
All listed relationships except for Diana Allers count for Paramour.
#Male Shepard
Ashley
Liara
Tali #
Miranda #%
Steve Cortez ^%
Kaidan ^
Jack #%
Diana Allers %$
#Female Shepard
Kaidan
Liara ^
Garrus #
Samantha Traynor ^%
Diana Allers ^%$