In Islam, what is the correct title ?

An Imam teaches in a Mosque and a Muezzin does the call to prayer.

You're Welcome.

Qaree teaches Quran in mosque.

Imam leads the prayer in a Mosque.

Moazin does the call to prayer.

peace Nemesis,

the prayer leader at a mosque is usually called an imam. scholarly/elderly persons are often given the title sheikh out of respect.

it might benefit you to know that there is judeo-christian equivalent of the clergy (or priestly class) in islam. thus, an imam is but a person chosen by the community to lead prayers based on this person's knowledge and character. in fact in principle any muslim can lead a prayer and he would be called the imam for that duration. similarly no priests/rabbis are required to bless childbirth, sanctify marriages, pray over the dead, prepare ritual food, etc. according to islam each muslim has a direct relationship with god without intermediaries.

god warns muslims in the qur'an of exaggerating in glorifying the prophets and saints:

(009:031)

They (certain Jews and Christians) take their priests and their anchorites to be their lords in derogation of Allah, and (they take as their Lord) Christ the son of Mary; yet they were commanded to worship but One Allah: there is no god but He. Praise and glory to Him: (Far is He) from having the partners they associate (with Him).

the person who makes the call for prayer (makes the "azaan") is called the muezzin.

hope that helps

The teacher could be an Iman, Sheikh, or Moulana.