What is the difference between the Torah and the Old testament?

The so-called "Old Testament" is a Christian reference to the Jewish sacred scriptures, referred to by Jews by its acronym as TaNaKh.  
In the Jewish tradition, this consists of three groups of books: 1) the "Ta" stands for Torah, the first five "Books of Moses" - Genesis, Exodus, Leviticus, Numbers, and Deuteronomy.  They speak of the history of humanity beginning with creation, continuing through the enslavement of the Children of Israel in Egypt and God's setting us free, the Revelation at Sinai ("the Ten Commandments") and the construction of the Tabernacle (all this in the first two books), the laws of holiness in the Temple and in the Israelite home (Leviticus), the forty years in the desert (Numbers), and the collection of Moses' speeches (Deuteronomy).
The second division, "Na", stands for "nevi'im" (prophets), and consists of the history of the children of Israel from the time they entered the Land of Israel until their return after the Babylonian exile (four books: Joshua, Judges, Samuel, and Kings), and of the prophecies that were found to carry meaningful messages beyond their immediate times (four books: Isaiah, Jeremiah, Ezekiel, and the Twelve "minor prophets" - minor in the sense that they left fewer writings).
The third division, "Kh", stands for sacred writings, and consists of eleven books. Since you asked only about Torah, I won't elaborate, for the sake of brevity...