Parashah is the weekly Scripture portion taken from the Torah. Each parashah is given a name and is usually referred to as "parashat - name" (e.g., parashat Noach). For more information about weekly readings, click here.
Aliyot refer to a smaller sections of the weekly parashah that are assigned to people of the congregation for public reading during the Torah Reading service. In most congregations it is customary for the person "called up" to recite a blessing for the Torah before and after the assigned section is recited by the cantor. For Shabbat services, there are seven aliyot (and a concluding portion called a maftir). The person who is called to make aliyah is referred to as an oleh (olah, if female).
Maftir refers to the last Torah aliyah of the Torah chanting service (normally a brief repetition of the 7th aliyah, though on holidays the Maftir portion usually focuses on the Holiday as described in the Torah). The person who recites the Maftir blessing also recites the blessing over the Haftarah portion.
Haftarah refers to an additional portion from the Nevi'im (Prophets) read after the weekly Torah portion. The person who made the maftir blessing also recites the blessing for the Haftarah, and may even read the Haftarah before the congregation.
Brit Chadashah refers to New Testament readings which are added to the traditional Torah Reading cycle. Often blessings over the Brit Chadashah are recited before and after the readings.
Mei Ketuvim refers to a portion read from the Ketuvim, or writings in the Tanakh. Readings from the Ketuvim are usually reserved for Jewish holidays at the synagogue.
Perek Yomi Tehillim refers to the daily portion of psalms (mizmorim) recited so that the entire book of Psalms (Tehillim) is read through in a month. For a schedule, of daily Psalm readings, click here.
Gelilah refers to the tying up and covering the Sefer Torah (Torah Scroll) as an honor in the synagogue.
Divrei Torah ("words of Torah") refers to a commentary, a sermon, or devotional on the Torah portion of the week.