The Torah is the Hebrew name for the first 5 books of the Bible. However, by extension, it has come to mean the entirety of Jewish teachings, and in modern Hebrew is even used in general to describe the total knowledge of any field.

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In its narrowest sense, Torah the first five books of the Bible: Genesis, Exodus, Leviticus, Numbers and Deuteronomy, sometimes called the Pentateuch. In its broadest sense, Torah is the entire body of Jewish teachings.
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Given the above, there is really no significant difference between saying that you're studying the Bible or the Torah. In Israeli schools, the study of scripture is called Tanakh, which is an acronym for Torah, Nevi'im, Ktuvim, where Nevi'im is "prophets" and Ktuvim (literally "the Writings") refers to the last set of books of the Bible. Together, Tanakh is the Hebrew term for the Bible (see e.g. http://www.jewishvirtuallibrary.org/jsource/Bible/jpstoc.html ).

As for the relationship between the Torah and the Old Testament, they are related, but the former is a part of the Jewish Bible, the Tanakh, and the latter is a modification of the Tanakh. The former was written in Hebrew while the latter was translated into Greek and then from Greek to e.g. English.

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All Old Testament canons are related to the Jewish Bible Canon (Tanakh), but with variations. The most important of these variations is a change to the order of the books: the Hebrew Bible ends with the Book of Chronicles, which describes Israel restored to the Promised Land and the Temple restored in Jerusalem; in the Hebrew Bible God's purpose is thus fulfilled and the divine history is at an end, according to Dispensationalism. In the Christian Old Testament the Book of Malachi is placed last, so that a prophecy of the coming of the Messiah leads into the birth of the Christ in the Gospel of Matthew.

The Tanakh is written in Biblical Hebrew and Biblical Aramaic, and is therefore also known as the Hebrew Bible (the text of the Jewish Bible is called the Masoretic, after the medieval Jewish rabbis who compiled it). The Masoretic Text (i.e. the Hebrew text revered by medieval and modern Jews) is only one of several versions of the original scriptures of ancient Judaism, and no manuscripts of that hypothetical original text exist. In the last few centuries before Christ Jewish scholars produced a translation of their scriptures in Greek, the common language of the Eastern portion of the Roman Empire since the conquests of Alexander the Great. This translation, known as the Septuagint, forms the basis of the Orthodox and some other Eastern Old Testaments. The Old Testaments of the Western branches of Christianity were originally based on a Latin translation of the Septuagint known as the Vetus Latina, this was replaced by Jerome's Vulgate, which continues to be highly respected in the Catholic Church, but Protestant churches generally follow translations of a scholarly reference known as the Biblia Hebraica Stuttgartensia. In 1943, Pope Pius XII issued the Divino Afflante Spiritu which allows Catholic translations from texts other than the Vulgate, notably in English the New American Bible.
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