I’ve written about this before, but for the sake of clarity:
TORAH — The first (5) books of the Bible, in every Biblical tradition.
The (5) books are:
In synagogues, one section from one of the books of Torah is read each Shabbat, in sequence, so that the entire Torah can be read in a single year, every year.
Each section is called a “parshah” or “sidrah.” Each parshah or sidrah has its own title, usually derived from a word in the first sentence. Each Shabbat has its own name, usually based on what parshah/sidrah is being read that day — e.g. “Shabbat Noach,” when parshah “Noach,” about Noah, is read.
“Kriyah” or “Kriyat ha-Torah” — The public recitation of Torah aloud to a congregation during Sabbath, Holiday and in Mon-/Thurs-morning services.
“Pentateuch” is a Greek word, meaning “‘five [penta] vessels,’ ‘five containers,’ or ‘five-volumed book’.” 
It designates “Torah.” It’s not commonly used in Judaism at this time, but was more common in previous generations. Rabbi J.H. Hertz’ chumash, for example, was entitled “Pentateuch and Haftarahs.”
Samaritan Torah — Another “Torah” tradition is that of the Samaritans (sometimes called “The Samaritan Pentateuch,” although “Samaritan Torah” would be more apt). It is their only actual scripture, although their tradition contains other “religious” books. Although there are around 6000 variations in it from the Masoretic Torah, these are considered minor, for the most part.
TaNaCH — The entire Jewish Scripture is made up of three major sections:
1. T = Torah
2. N = N’vi’im — “Prophets.” “Navi” means “prophet;” “n’vi’im” is the plural form of “navi.” There are 15 “literary” prophets — prophets who wrote “books.” Yishiyahu (Isaiah), Yirmiyahu (Jeremiah) and Yehezkiel (Ezekiel) are called the “major” prophets, because they wrote the (3) longest books. The remaining (12) are called “minor” prophets, because their books are smaller, not because they’re less important. There are other prophets whose name we know, who didn’t write books: Eliyahu (Elijah), Elisha, Natan, and so on. There are even more prophets mentioned, whose names we don’t know at all. Most haftarahs read on Shabbat and holidays are comprised of chapters from one of the n’vi’im.
3. CH = K’tuv’im — “Writings.” The first Hebrew letter of this word is [כ] — “Kaph” — which is more or less parallel to the English letter [K]. However, at the end of a word, this letter can be pronounced “ch.” So, the initial letters of these (3) sections form the acronym “TaNaCH,” with the vowel (a) added to make the three consonants pronounceable. “K’tuv’im” includes non-prophetic books that are still considered to be Divinely inspired. “Tehillim/Psalms” is a collection of song lyrics (that look like poems). “Mishlei/Proverbs” is a collection of wise sayings, usually about a single sentence long. There are also historical books (e.g. Shmuel/Samuel I & II), “Ruth,” “Esther,” that are more prose-like in form.
Chumash — “Chumash” means (5) in Hebrew. A “chumash” is a book containing Torah — “The 5 Books of Moses” — separated into its weekly readings. A chumash can also contain a commentary (e.g. the “Hertz” chumash, with the commentary of Rabbi J.H. Hertz), and can also include the haftarahs associated with each parshah. A “chumash” is not the entire Jewish scripture, although it’s the main tool by which Torah can be studied.
Talmud — Understood to be the “oral” tradition of understanding that accompanied the “written” Torah. Jews differ on what place they give the Talmud, but it can always be a valuable resource in innumerable ways. The Talmud was originally taught orally, but was later written down, itself.
The Talmud primarily consists of:
1. The Mishnah — the application of the laws of Torah
2. The G’morah — the discussion and debate surrounding the Mishnah.
Testament or Covenant — An “agreement,” especially a legal agreement specifying the duties of both parties to the agreement. “Testament” might additionally imply a written agreement or contract. “Torah” is largely the “contract” between G-d and the people of Israel, although it has implications for the entire world.
“Old Testament” — The prophet Yirmiyahu/Jeremiah speaks of a “new covenant” that will be written on people’s hearts.  The Jewish under-standing of this is that this “new agreement” doesn’t supersede the details of the “old” one; it simply adds an extra spiritual dimension to it.
“New Testament” — Literarily, the “New Testament” includes:
1. The (4) Gospels (the narratives of the life and teachings of Jesus)
2. The book of “Acts,” which details the actions of Jesus immediate followers, the Apostles, after the events in the Gospels
3. “Epistles” — letters written by several of the Apostles, mostly “Paul”
4. “Revelations” — a visionary book about the future.
The “New Testament” is not the entire Christian Bible per se; it’s part of it.
These books are not accepted as part of the Jewish Scriptures [“Messianic Judaism” notwithstanding], although the Jewish Scriptures comprise a large part of the Bible in the various Christian traditions.
In Christian belief, G-d cancelled His contract with the Jews and replaced it with a “new contract,” (see note  for the source of the phrase), based on the teachings and person of Jesus. In Jewish belief, G-d’s agreement with the Jews was/is eternal; if we don’t keep our part, we can and should repent and the positive benefits of the contract will be reinstated. Thus, “Old Testament” and “New Testament” aren’t terms that Jews can use to accurately indicate Jewish and Christian Scriptures, although often used that way popularly because of the wider culture.
There are multiple “biblical” traditions. The “Jewish” tradition is the oldest, although it once contained books that are no longer considered “canonical,” and might have had textual variations at one time. The Septuagint — the first translation of Jewish books into Greek — includes books that are part of “Jewish” tradition (called the “Masorah” or “Masoretic” tradition), plus some others which are included in Catholic and other traditions (e.g. “ben Sirach;” “Judith;” etc.), but not in Jewish Scriptures (although written by Jews) and not in Protestant tradition (in which these books are called “Apocrypha”).
In all churches, a section from Scripture is read in Sunday/Sabbath services (sometimes more than one section is read). This public reading of Scripture is based on Jewish practice, even if the selections themselves differ in content. 
As mentioned above, the first (5) books in all traditions are the same, and comprise “Torah.” After that, each tradition includes various books — some the same; some not — and arranges them in various orders.
As stated above, the public reading of Torah in synagogue is called the “kriyah” (“reading aloud” or “recitation”) or “kriyat ha-Torah” (“recitation of the Torah”).
This is also the root of “Qur’an” — “The Quran (…literally meaning “the recitation”, also romanised ‘Qur’an’ or ‘Koran’) is the central religious text of Islam…” 
The Qur’an is read publicly in Muslim services. This, as with the Christian reading of Scripture, is based on Jewish practice. 
While Torah was divided by the rabbis into parshahs or sidrahs, the Qur’an is divided into “surahs,” each of which has a title (usually) based on a word in the first sentence or two. Some scholars have suggested that “surah” was originally derived from “sidrah”: “The very word ‘surah’ owes it’s existence, as I believe…to ‘sidrah’.” 
Does this help clarify some of the similarities and differences?
 Yirmiyahu/Jeremiah 31:31-34
 “For the law of Moses has been preached in every city from the earliest times and is read in the synagogues on every Sabbath.” (Acts 15:21)
“The first converts to Christianity (being either Jews…) were personally familiar with the liturgical customs of the synagogue. In fact, the earliest Christians continued to participate in synagogue worship as long as they were permitted, and some Christians (e.g., Paul) even carried out a teaching ministry in the synagogue. It is not surprising, therefore, that the basic pattern and elements of Christian worship came from the synagogue service. Nowhere is this clearer than in the reading and preaching of Scripture in worship.”
 The topic of the Jewish roots of Islam is difficult to discuss these days, although the evidence is there. Naming their Scripture “Qur’an” is as if to say: “This recitation will replace the public recitations of the Jews (Torah) or Christians (Gospels, etc.),” which is in keeping with the Muslim teaching that Islam supersedes Judaism and Christianity. However, we can appreciate the evolution of the form — public recitation of Scripture — that began with Jewish practice. Nor do “the Jewish roots of Islam” negate Islam’s own inspired quality. Rembrandt didn’t invent oil painting; Einstein didn’t invent mathematics; Muhammad didn’t invent the Arabic language. Each were inspired to use materials that already existed to express something unique.
 Abrahams, Israel; Jewish Quarterly Review, Oct. 1907; vol. XX, no. 77; p. 877