There are many many places to begin talking about Judaism and Jewish people, but as a "Religion of the Book" then a good place is with Torah. The word itself can mean several things,
As a noun, not as a proper noun, the Hebrew word "torah" most accurately means "instruction" or "doctrine" in Hebrew. However, it is also often translated as "law" but the word is related to the verbs "to teach" or "to instruct" and not "to legislate" instead. Although we are discussing the word as a proper noun, it is helpful to understand its etymological background.
Five Books of Law
The word "Torah" normally refers to the five books of Bereshit (Genesis), Shemot (Exodus), Vayikra (Leviticus), Bamidbar (Numbers), and Devarim (Deuteronomy). These five books are known in Greek as the Pentateuch. This is the "default" usage of the term in general.
Torah and Talmud
Traditionally, the word "Torah" is also used as a synecdoche for both the Torah and the Talmud, which are referred to as the תורה שבכתב torah she-bi-khitav (written Torah) and the תורה שבעל פה torah she-beat peh (spoken Torah). If someone asks if something is "according to Torah" then they may receive an answer that cites either the five books of the written Torah, or the six tractates of the oral Torah as written in the Talmud.
At its broadest, as all Jewish law is based on the five books of the Torah, the word Torah can be used to refer to Jewish law in general. If someone says "I will devote myself to studying Torah" then this could mean they are studying Torah and Talmud in particular, but also many other writings which involve Jewish law.
Torah is one part of the Tanakh, which is the canonized Jewish scripture. The Tanakh is roughly equivalent to the Old Testament in the Christian canon. The components of the Tanakh and the books within them are,
The Torah comes in one of two formats: in a formal scroll, appropriate for highly regulated use generally in synagogues only; or in less-formal forms, such as paginated books, which are appropriate for more regular use and study. These two formats have their own terms,
ספר תורה Sefer Torah
A Sefer Torah is an extremely formal item: a parchment scroll, containing the entire of the five books of the Torah, and written by hand. Each letter and accompanying diacritics must be totally precise. It takes months or years for a specially trained scribe — called a sofer — to prepare a Sefer Torah from start to finish. A Sefer Torah must be handled with the highest possible respect, and it is forbidden to touch it directly. The scroll may be unrolled and read, but after normal wear and tear it must be replaced if even a single letter is damaged.
Any version of the five books of the Torah that is not a Sefer Torah — e.g., a printed book — is known as a חומש humash. This comes from the Hebrew word חמש hamesh meaning "five" and refers to the five books. A more formal term than חומש Humash is חמשה חומשי תורה Hamishah Humshei Torah, meaning "five fifths of Torah" to reflect both its individual components and its unified whole.