A parashah (plural: parashot or parashiyyot) is a section of a biblical book in the masoretic text of the Tanakh (Hebrew Bible). Such sections are designated by various types of spacing between them in Torah scrolls, in scrolls of the books of Nevi'im or Ketuvim (especially megillot), and in masoretic codices. The division of the text into parashot for the biblical books is independent of chapter and verse numbers, which are not part of the masoretic tradition. Parashot are not numbered, but some of them have special titles.
Incorrect division of the text into parashot, either by indicating a parashah in the wrong place or by using the wrong spacing technique, halakhically invalidates a Torah scroll according to Maimonides.

Rationale of the section divisions
In most modern scrolls and Jewish editions of the Bible, there are two types of parashot: An "open portion" (parashah petuhah) and a "closed portion" (parashah setumah). An "open portion" is roughly similar to a modern paragraph: The text of the previous portion ends before the end of the column (leaving a space at the end of the line), and the new "open" portion starts at the beginning of the next line (but with no indentation). A "closed portion", on the other hand, leaves a space in the middle of the line of text, where the previous portion ends before the space, and the next portion starts after it, towards the end of the line of text.
An "open portion" (petuhah) is often abbreviated with the Hebrew letter "פ" (peh), and a "closed portion" (setumah) with the Hebrew letter "ס" (samekh).
In masoretic codices and in medieval scrolls, these two spacing techniques allow for a larger range of options:

An "open portion" always started at the beginning of a new line. This could happen the way described above, but also by leaving a blank line between the two portions, thus allowing the previous to sometimes entirely fill its last line of text.
A "closed portion" never began at the beginning of a line. This could happen as in modern scrolls (a space in the middle of a line), but also by the previous portion ending before the end of the line, and the new portion beginning on the next line after an indentation. Spacing techniques
According to the ruling of Maimonides (Laws of Tefillin, Mezuzah and Torah Scrolls 10:1), any error regarding a parashah completely invalidates a Torah scroll. This includes a parashah in the wrong place, of the wrong type, or a missing parashah.
However, there is also a responsum by Maimonides
All of them rule that a scroll containing parashot based on alternative scribal traditions that disagree with Maimonides' list of parashot is nevertheless a valid scroll. However, a blatant error with no source in any scribal tradition does invalidate a Torah scroll, even according to the lenient opinion.

Halakhic significance
The following list presents the parashah divisions as found in Torah scrolls, in Maimonides' Mishneh Torah, and in the Aleppo Codex (the very few inconsistencies between these three sources are explained in footnotes). The five books of the Torah have been broken down into their weekly Torah readings for convenience.
Due to the influence of Maimonides, the parashah divisions in the Torah have been highly standardized, and therefore the list below conforms to the section divisions as found in nearly all printed Jewish bibles in modern times, in Torah scrolls, as well as in similar online texts. Variations found in alternative masoretic traditions (such as in the Leningrad Codex) are provided in a separate list.
Symbols:
Note: Only breaks between two sections are listed: Any open or closed break between two sections indicates the status of the second one that begins with it, e.g. in Genesis {S} 5:32-6:4 {P} is considered a "closed" section (setumah) because it begins with {S}. No section break is indicated before the first portion of a biblical book, or after its last portion.

{P} = parashah petuhah ("open portion"), typically resembles a new paragraph
{S} = parashah setumah ("closed portion"), typically represented as a blank space in the middle of a line
{-} = no parashah break indicated Parashot in the Torah
The full Hebrew letter-text of Genesis with parashah divisions may be viewed here.

Parashat Bereshit (Genesis 1:1-6:8): 1:1-5 {P} 1:6-8 {P} 1:9-13 {P} 1:14-19 {P} 1:20-23 {P} 1:25-31 {P} 2:1-3 {P} 2:4-3:15 {S} 3:16 {S} 3:17-21 {P} 3:22-24 {P} 4:1-26 {S} 5:1-5 {S} 5:6-8 {S} 5:9-11 {S} 5:12-14 {S} 5:15-17 {S} 5:18-20 {S} 5:21-24 {S} 5:25-27 {S} 5:28-31 {S} 5:32-6:4 {P} 6:5-8
Parashat Noach (Genesis 6:9-11:32): {P} 6:9-12 {S} 6:13-8:14 {S} 8:15-9:7 {S} 9:8-17 {P} 9:18-29 {P} 10:1-14 {S} 10:15-20 {S} 10:21-32 {P} 11:1-9 {P} 11:10-11 {S} 11:12-13 {S} 11:14-15 {S} 11:16-17 {S} 11:18-19 {S} 11:20-21 {S} 11:22-23 {S} 11:24-25 {S} 11:26-32
Parashat Lekh Lekha (Genesis 12:1-17:27): {P} 12:1-9 {P} Genesis

Parashat Ki Tissa (Exodus 30:11-34:35): {P} 30:11-16 {P} 30:17-21 {P} 30:22-33 {S} 30:34-38 {S} 31:1-11 {P} 31:12-17 {S} 31:18;32:1-6 {P} 32:7-14 {S} 32:15-35 {S} 33:1-11 {P} 33:12-16 {P} 33:17-23 {P} 34:1-36 {P} 34:27-35
Parashat Vayakhel (Exodus 35:1-38:20): {S} 35:1-3 Exodus
The full Hebrew letter-text of Leviticus with parashah divisions may be viewed here.

Parashat Acharei Mot (Leviticus 16:1-18:30): {S} 16:1-34 {P} 17:1-16 {P} 18:1-5 {S} 18:6 {S} 18:7 {S} 18:8 {S} 18:9 {S} 18:10 {S} 18:11 {S} 18:12 {S} 18:13 {S} 18:14 {S} 18:15 {S} 18:16 {S} 18:17-30
Parashat Kedoshim (Leviticus 19:1-20:27): {P} 19:1-22 {P} 19:23-32 {S} 19:33-37 {P} 20:1-27
Parashat Emor (Leviticus 21:1-24:23): {P} 21:1-9 {S} 21:10-15 {S} 21:16-24 {P} 22:1-16 {P} 22:17-25 {S} 22:26-33 {P} 23:1-3 {P} 23:4-4 {P} 23:9-14 {S} 23:15-22 {P} 23:23-25 {S} 23:26-32 {P} 23:33-44 {P} 24:1-4 {P} 24:5-9 {S} 24:10-12 {P} 24:13-23
Parashat Behar (Leviticus 25:1-26:2): {P} 25:1-7 {S} 25:8-24 {S} 25:25-28 {S} 25:29-34 {S} 25:35-38 {S} 25:39-46 {S} 25:47-26:2
Parashat Bechukotai (Leviticus 26:3-27:34): {P} 26:3-13 {P} 26:14-26 {S} 26:27-46 {P} 27:1-8 {S} 27:9-34 Numbers
The full Hebrew letter-text of Deuteronomy with parashah divisions may be viewed here.

Parashat Ki Tavo (Deuteronomy 26:1-29:8): {P} 26:1-11 {S} 26:12-15 {S} 26:16-19 {P} 27:1-8 {S} 27:9-10 {S} 27:11-14 {S} 27:15 {S} 27:16 {S} 27:17 {S} 27:18 {S} 27:19 {-} 27:20 {S} 27:21 {S} 27:22 {S} 27:23 {S} 27:24 {S} 27:25 {S} 27:26 {P} 28:1-14 {P} 28:15-68 {S} 28:69 {P} 29:1-8
Parashat Nitzavim (Deuteronomy 29:9-30:20): {P} 29:9-29:28 {S} Variants
Works cited in the references to this article:

Ofer, Yosef. "The Aleppo Codex and the Bible of R. Shalom Shachna Yelin" in Rabbi Mordechai Breuer Festschrift: Collected Papers in Jewish Studies, ed. M. Bar-Asher, 1:295-353. Jerusalem, 1992 (Hebrew).
Penkower, Jordan S. "Maimonides and the Aleppo Codex". Textus 9 (1981):39-128.
Penkower, Jordan S. New Evidence for the Pentateuch Text in the Aleppo Codex. Bar-Ilan University Press: Ramat Gan, 1992 (Hebrew).