The above resources are indeed excellent surveys of the issues surrounding textual criticism of the bible. A couple of extra points are worth noting:
(1) The question asks about the textual history of the Bible. The Greek manuscript families relate specifically to the New Testament. If the question includes the Old Testament that would need to be looked at as a separate question.
(2) It is true that modern translations give greater weight to the Alexandrian family of texts. But a more accurate description would be to say that modern translations use an eclectic text. The leading scholarly Greek NT text is that published by the United Bible Societies. This text is the best current scholarly recreation of the original text. It is based on the study of many manuscripts, lectionaries, commentaries and ancient translations. Where there are variant readings the translators make their best call on a case by case basis as to which reading is right. The resources already mentioned in the above comments will set out the principles involved. No one manuscript source is considered perfectly reliable, so the UBS Greek text will not match any specific manuscript or family of manuscripts.
(3) As regards the text used by particular translations, the introduction to most modern translations will include a statement on which text they have used and why. For instance the 2011 edition of the NIV includes these paragraphs:
For the Old Testament the standard Hebrew text, the Masoretic Text as published in the latest edition of Biblia Hebraica, has been used throughout. The Masoretic Text tradition contains marginal notations that offer variant readings. These have sometimes been followed instead of the text itself. Because such instances involve variants within the Masoretic tradition, they have not been indicated in the textual notes. In a few cases, words in the basic consonantal text have been divided differently than in the Masoretic Text. Such cases are usually indicated in the textual footnotes. The Dead Sea Scrolls contain biblical texts that represent an earlier stage of the transmission of the Hebrew text. They have been consulted, as have been the Samaritan Pentateuch and the ancient scribal traditions concerning deliberate textual changes. The translators also consulted the more important early versions — the Greek Septuagint, Aquila, Symmachus and Theodotion, the Latin Vulgate, the Syriac Peshitta, the Aramaic Targums, and for the Psalms, the Juxta Hebraica of Jerome. Readings from these versions, the Dead Sea Scrolls and the scribal traditions were occasionally followed where the Masoretic Text seemed doubtful and where accepted principles of textual criticism showed that one or more of these textual witnesses appeared to provide the correct reading. In rare cases, the committee has emended the Hebrew text where it appears to have become corrupted at an even earlier stage of its transmission. These departures from the Masoretic Text are also indicated in the textual footnotes. Sometimes the vowel indicators (which are later additions to the basic consonantal text) found in the Masoretic Text did not, in the judgment of the committee, represent the correct vowels for the original text. Accordingly, some words have been read with a different set of vowels. These instances are usually not indicated in the footnotes.
The Greek text used in translating the New Testament is an eclectic one, based on the latest editions of the Nestle-Aland/United Bible Societies’ Greek New Testament. The committee has made its choices among the variant readings in accordance with widely accepted principles of New Testament textual criticism. Footnotes call attention to places where uncertainty remains. 2