Coloratura: The soprano with the highest (usually) range, has a lot of staccato trills and Ahhh-ing. If you want to hear a classic coloratura aria, listen to "Der Holle Rache" from Die Zauberflote, or "Les Oiseaux dans la Charmille" from Les Contes d'hoffmann. It's light and acrobatic. Uusally minimal lower register.
Lyric: The heroine/ingenue characters, like Violetta, is a graceful voice, with more legatto lines, very strong upper register, and good lower. Liu, Manon, Mimi, etc are lyrics.
Dramatic: Not necessarily the Wagner roles (those are Wagnerian sopranos), these obviously have the most dramatic, rich sound of the sopranos. Their range usually doesn't go as high, but they have a powerful bottom register. Think Tosca, Turandot, Butterfly to an extent.
Spinto: A full lyric soprano that can be pushed to dramatic climaxes. Butterfly, Aida, Donna Elvira. Tosca might be considered a spinto as well.
Usually, you can be more than one voice type, for example, you can sing Tosca, a dramatic role, but also sing Violetta, a lyric who must sing coluratura for the fist act of La Traviata. Usually the coloraturas have the hardest time switching because they lack some of the richness and bottom register. However, people like justagirl2 can be a dramatic coloratura. (Usually, you are never JUST a spinto. A spinto refers more to the role than to the voice.)
So, if your teacher can tell you're a coloratura already (around how old are you?) than that's great, but if you want to work on belting, than I would really suggest working your lower register more. (But a coloratura doesn't have to be a soprano, there are coloratura Mezzo role like Rosina)
Hope this helps!
~And let us try, before we die, to make some sense of life~