(For People who don't know their Greek and Hebrew)
The original language clarifies a lot of Scriptures which can be ambiguous in English, and at the same time, the English translation can be interpretive or assumptive when the original language is non-specific. There are many instances where a serious Bible student will want to do some language checks.
Webster's Colligate Dictionary - To be fair, this (or another English dictionary) is an important reference. All the archaic words used by the KJV are covered in the dictionary, and nobody has a perfectly complete vocabulary. Honestly, I opened my dictionary in Bible study more often as a young man than I do now, but English is before Greek and Hebrew...
Strong's Exhaustive Concordance of the Bible - This is the de facto standard for key-word studies. In the front section of the book, you look up any English word, and it lists all the Bible verses including that word (keyed to KJV), and by each verse reference is a number for a Greek or Hebrew (or Aramaic) word definition. The back section of the book has the definitions listed in numberic order for simple lookup. The word definitions are well written, and reference root words and similar words. Between the definitions and understanding the roots and similar usages, the reader gets a good understanding of the word used in the original language.
As the long-established standard, 'Strongs Numbers' are used in many reference books and software, and are even shown in superscipt in many electronic Bible texts. Even though the English wordlist to find these numbers are keyed to KJV, modern references use the number with whatever version they choose, as they represent the original language words. Strong's numbers and definitions are almost always included (free) in any Bible software, and is available on online resouces. The physical book is alarmingly large (second only to my unabridged dictionary), but inexpensive.
Vine's Expository Dictionary of Biblical Words - This is the long-time standard companion reference for Strongs. It is divided into Old and New Testaments, you look up the word in English, and it gives you expanded definitions of original language words which correspond to that English word. Simple, useful, and more depth of understanding than using Strongs by itself. For basic key-word studies Strongs and Vines make an excellent team. Like Strongs, Vine's is long out of copyright, and so is freely available in electronic references, and the physical book is available and inexpensive.
Parallel/Interlinear Bibles give you the text of multiple translations side-by-side (or above/below) so that you can directly compare. Anyone can benefit from seeing the literal word-for-words, but Interlinears are really nice if you actually have had 'some' Greek/Hebrew, or are currently taking a biblical language course. There are several distinctions regarding source-texts which must be understood when using these interlinears.
Old Testament - Masoretic Text The Masoretic text is used for the KJV and is the primary source for almost all modern versions, including the NASB, ESV, NIV, and many others. Green's Interlinear Bible uses the Masoretic Text.
New Testament - Textus Receptus Text The Textus Receptus (TR) is used for the King James Version of the Bible, and for the majority of the text of the New King James Version. Green's Interlinear uses the TR for the New Testament.
English Language - Green's Literal Translation of the Bible The English words used below each original language word are literal translations of the word they are under. In side columns are Green's "Literal Translation of the Bible" which are not exactly the same. For example, if you read interlinearly from Genesis 4:14, you get, "And said God, Let be luminaries...," while in the column you will read, "And God said let luminaries be...".
Old Testament - Septuagint Text The Septuagint text is preferred by Eastern Orthodox churches and is the version which Christ and the Apostles quoted in the New Testament. Brenton's Septuagint with Apocrypha uses this text. This is a parallel presentation in Greek with outside columnes of literally translated English text. Note that this Bible is Old Testament only.
New Testament - Critical Texts The Critical Texts (generically) vary from the TR and is the Greek source for the New Testament of almost all modern Bibles. The Interlinear Bible I recommend is Alfred Marshall's Interlinear NASB-NIV Parallel New Testament in Greek and English which uses the 21st ed of Nestle's Novum Testamentum Graece, which was the conservative standard at the time of publication (1993). Like Green's Interlinear, this Bible has the Greek in the center, with literal translation below each word, but the side columns are NASB on the left and NIV on the right. Note that this Bible is New Testament only.
There are two references which I use, both of which I think I can call modern standards. At least I can say that they are popular in my circles, and both are still available new, since my purchases in the late 1990's. These are references where the authors give actual commentary about the translation, and possible alternative translations.
Expositor's Bible Commentary by Frank E Gaebelein - The set is comprised of commentary by several authors, but the publication is done to a consistantly high standard. These commentaries bring a rather technical discussion on passages, often including aspects of translation.
New English Translation (NET) Bible - The NET Bible is uniquely valuable for it's incredibly extensive translation notes, which are well written and meaningful, explaining why the word or passage was translated as it was, and even going into alternate considerations. The extensiveness of these notes make this more of a translation reference than a Bible version, and I am not aware of any similar reference works.