Fach is a word used in opera to determine vocal type - and in determining vocal type, determining operatic roles sung by that type or fach.
Type in theatre is very similar but is has a double edge in that it is not just voice type (music theatre) but also physical type.
Ironically, the physical type is becoming more and more prominent in opera too...ANOTHER blog!
Theatre/Music Theatre types can be some of the following: (the ideas are endless really)
girl next door type
leading lady/man type
femme fatale type
young mother type
shy plain jane type
and on and on - it is endless, and obviously has more variety and change than opera.
Some casting people will also associate your "type" with a performer that has defined themselves often with a specific role - "Kristin Chenoweth type" or "Bernadette Peters type"
Obviously these terms work for straight theatre as well as music theatre, thus the physical types and vocal types grouped together.
Ultimately, as a singer, it would be nice to be able to match your physical type with your voice type, (ingenue/soubrette voice) but it doesn't always work out that way. So, sometimes, we as performers have to create that type that best reflects us so casting people don't have to get creative. There is no time in the audition process to get creative anyway.
YOU must decide HOW you want to be seen, and how your voice reflects what is seen. We see you first - therefore whatever "type" you exude must be in that presentation. When you open your mouth sing, it needs to further convince them of that.
Types basically come from the mythical sense of archetype. Research that. You will find all theatrical types clearly develop from that. Voices are associated with archetypes, and thus the distinction. Don't dream of what you would like to be, but rather, BE what you ARE. Find an honesty in discovering your type and develop it. As I have said in an earlier missive, we need ALL KINDS of characters. Embrace yours! If you aren't sure, work with voice and acting teachers to discover fully what you have to offer.
Types in theatre have many variances and shades. Discover a primary one and create the shades and hues around that type. This will keep you from putting yourself in a box. Rather, when you walk into that audition room, you can present the type they need to see and hear in order to be hired for the job, or at least, given a second glance - and maybe a callback!