I'm sure I'll miss some important things, but here's a list of what I'd consider essential (or at least very useful). I'm going to write for the English speaker, since that's what I'm used to. I'm community-wikifying this answer, so anyone with 100 reputation points can edit it. I'm going to type this out without links at first and come back in and linkify it later. Others are welcome to help with the linkifying.
ESSENTIAL
- Torah (Five books of Moses) with Rashi's commentary and translation of both the text and commentary. Torah is our central text. Rashi is the universally-acknowledged dean of commentators. We're required to study the weekly portion with commentary, and this would facilitate that.
- Other commentaries on the Torah for added variety or depth of study. Mikraot Gedolot or Torat Chaim provide a nice collection of commentaries. I'm partial to the translation and commentary of R' S. R. Hirsch.
- Tanach with translation for reference.
- Tanach with commentary (either collection of primary commentaries, such as Mikraot Gedolot or an English commentary) for study.
- Mishnah with standard commentaries or Kehati
- Talmud
- All of these dictionaries:
- Modern Hebrew - Helpful with Tanach, Mishna, Hebrew commentaries, and Rabbinic literature
- Jastrow on Aramaic - Comprehensive coverage of pretty much any word you'll see in the Talmud and in Aramaic commentaries
- "Practical Talmud Dictionary" - Very helpful with understanding Talmud phrases in context
- Dictionary of acronyms / "Otzar Rashei Teivot" - Talmud commentaries, and Rabbinic literature are full of acronyms that can be frustrating if you don't have a reference handy. There are many brands; in my experience, any is fine.
- Distillation of Halacha to look up what the issues are (but not necessarily to tell you what to do in every case). At least one of the following, some or all of which can be had with English translation:
- Mishna Berura
- Kitzur Shulchan Aruch
- Aruch Hashulchan
- Shulchan Aruch Harav (if you're Chabad)
- Yalkut Yosef (if you're Sepharadi)
- Ben Ish Chai (if you're Sepharadi)
- Rambam's Yad Hachazaka - Whichever edition[s] suit your learning style and level, e.g.:
- With traditional commentaries
- "Rambam La'am"
- With English translation
- At least one thing not on this list, chosen based on your interests or desire for expansion on something in this list. I think it's essential to have books in your library that you chose, to increase your personal connection to the library and to learning, and to make learning more enjoyable and therefore more likely to be frequent.
OPTIONAL