Unicode is a standard “registry” of characters designed to catalogue all the world’s writing systems and serve as a superset of all previously-existing character sets used and supported by just about everyone and all their software. Get into the details on Stack Overflow or see some of the Unicode encodings on Wikipedia.
PDF charts on unicode.org — The only complete and perpetually up-to-date reference.
UNIDATA on unicode.org — “This directory contains the final data files for the Unicode Character Database, for Version 8.0.0 of the Unicode Standard.”
CopyPasteCharacter — Contains a bunch of common characters for easy copying.
Unicode planes reference — Confused about what “BMP” means? Check here.
codepoints.net — Powerful search engine with previews in Unifont
shapecatcher — Recognizes Unicode characters through a drawing field.
Kanji search on jisho.org — Find kanji characters by their parts.
Google Translate — Click the pencil icon in the input area to enable shape recognition. Great for writing short bits of text in a language you don’t know.
ASCII Xlate — Converts plain ASCII, binary, octal, hex, base32, base64, ASCII85, and decimal ASCII. Also calculates various hashes!
Unicate — Converts to various Latin Unicode “alphabets” (e.g. full-width, math scripts, etc.).
Text converter on qaz.wtf — Similar to the above.
Zalgo generator on eeemo.net — generates Zalgo text.
Convert Text on the Chrome web store — Converts the case of text as a Chrome extension. Includes Zalgo generator and fullwidth transform.
Acrostic generator — Dubiously useful. Also converts to Unicode Math Monospace.
Abbreviator — Saves characters in tweets by using precomposed characters.
Unitools — A good compilation of a bunch of other tools. Honorary mention.
Alan Wood’s Unicode font list — Probably the most complete list of high-coverage fonts.
PragmataPro — A monospaced programming font with 6,000 glyphs (and rising).
CharacterMap — Analyzes glyphs from font files.
Google Noto Fonts — A set of sans and serif fonts supporting 581 languages (as of April 2016), with about 50% glyph coverage.
Preview of all codepoints in the BMP — Useful for testing coverage.
Full Emoji Charts — all emoji with comparison pictures of implementations on various platforms.
Emoji Symbols: Background Data on unicode.org, a.k.a. (L2/09-027) — Japanese carrier background data for the original emoji import (mostly historic value).
List of emoji ZWJ ligatures on unicode.org — Note: These are not actually emoji or part of the Unicode standard. Implementation, support, and blame lies entirely with the third parties involved.
Text vs Emoji reference on unicode.org — Shows which characters (should) render as emoji or text.
ASCII art collection on asciiart.website, formerly chris.com
Vim — Insert mode:
<C-X> 8 <CR> xxxx <CR>
Mac OS X — (☑︎ Unicode Hex Input)
Windows — (☑︎ Registry Key)
Unix in GTK applications —