«Преподава́тель(ница)» is someone who teaches students in the university (it roughly overlaps with the English word 'professor'), «учи́тель(ница)» is someone who teaches people in school.
Преподавательница sounds a tad weird. (I know, dictionaries have this word but I guess you have to trust my word here. Or not :)). More generally, even though names of certain professions do have feminine forms, you need to exercise caution using them. Check the notes in the dictionary for разг./разговорное, прост./простонародное, уст./устаревшее and act accordingly.
The second point, it is possible to call a school teacher преподаватель. All it does, it makes your speech sound more formal. The opposite is not true. One cannot call a university professor учитель or учительница, these words are reserved exclusively for school teachers.
> guess you have to trust my word here. Or not
Being a native speaker, I would rather trust my own language intuition, sorry ^^'
-тель is a fairly common suffix to form a noun that names the object performing an action of a verb. Here are a few examples:
- учитель (the only with the plural in -я)
- выключатель, переключатель
- родитель (quite unusual due to formation from a perfective verb)
It is not the only one, and you should know whether the word exists anyway. The one who buys is покупатель but the one who sells is продавец.