Many books in the Bible are named after particular people, but readers should not assume that the title of the book accurately indicates the author. In the Old Testament, books are typically named for the main character rather than the author (e.g., Joshua, Esther, Jonah). There are references to the authorial activity of Moses, David, Solomon, Jeremiah, and Nehemiah, among others, but none of them is probably the final author of the books associated with them.
For example, the Pentateuch is traditionally ascribed to Moses, but since it describes his death, Moses can hardly be the author, at least of the final form of the book. There are many indications that the Pentateuch was written by several people over a long period of time. Stylistic differences among different sections and the repetition of similar material suggest multiple authorship (e.g., the two creation stories in Gen 1 and Gen 2, the two versions of the Ten Commandments in Exod 20 and Deut 5). Historical anachronisms (historical data true when the Pentateuch was written but not in Moses' day) suggest a long period of time over which the Pentateuch was written (e.g., Gen 12:6: "The Canaanites were then in the land," suggesting that when the verse was written, they were no longer in the land; Deut 1:1: "beyond the Jordan" as a reference to the east side of the Jordan, a perspective that implies residence on the west side of the Jordan; Deut 17:14-20: references to a king, when kings were apparently not anticipated so early in Israel's history).
In the New Testament, the authors of letters are usually named, though scholars debate the attribution of some of the letters (e.g., the Pastoral Epistles or 2 Peter). (The letter to the Hebrews is an exception, because no author is mentioned; though it is traditionally associated with Paul, the differences in style from other Pauline writings and the lack of Paul's name in the text itself argues against Pauline authorship.)
Unlike the letters, the gospels and Acts do not name their authors, although Luke and Acts have similar dedications to Theophilus at the beginning and thus presumably the same author. Early Christian traditions associated our canonical gospels with Matthew, Mark, Luke, and John. However, it is interesting to note that Papias, a prominent Christian writer from the early second century, said that he trusted oral reports from people who had known the apostles more than any written word, so he, at least, seems to have had some doubts about the apostolic authorship of the gospels.
Questions of date and authorship of the biblical books can be important matters for interpreting the biblical text, since knowing the proper historical context can aid in understanding otherwise ambiguous references. Good commentaries and introductions to the Old Testament and New Testament contain a wealth of information on questions of date and authorship, as well as other matters.
Before we can answer the question, "Who wrote the books of the Bible," we need to know more about how the books were composed. In many cases, ancient books do not have a single author, so questions are date and authorship are more complex than readers of the biblical text might assume.