Reading from the Torah during services, or reciting a blessing over such a reading. This is considered an honour, and is one of the primary rites of passage during a Bar Mitzvah.
The first aliyah of a day's reading is given to a kohein, the second to a Levite, and the subsequent aliyoth are given to people celebrating major life events, such as a marriage, a bar mitzvah, or the birth of a child.
Usage is a little odd; one makes aliyah, or is given an aliyah.
Pronounced ah-lee-AH. Plural is aliyoth.
As stated above, it can also refer to a person immigrating to Israel; it does not necessarily refer to a person belonging to the specific Aliyah movement mentioned above.