Word puzzle

Word Puzzles are puzzles revolving around words and wordplay, and are certainly one of the most common puzzle types (if not the most common puzzle type) found in puzzle hunts. This popularity and prevalence in the hunt community is likely due to both the accessibility of them (linguistics, particularly the English language, being a well-documented topic), and word puzzles' long history of public consumption, with some of the earliest word puzzles dating back to at least the first century AD.

Core Characteristics
As the name would suggest, word puzzles have a core topic that makes them word puzzles as opposed to any other type: words. If a puzzle's main conceit is identifying, manipulating, or placing words in a particular arrangement, they'd be considered a word puzzle. While there are some rare cases when a puzzle does deal with these concepts, but is not a word puzzle, most will at least end up being partially word puzzles.

To differentiate between a word puzzle and another type of puzzle that happens to involve letters or linguistics, look at what you're actually doing with the words, and whether or not replacing the linguistic theme with another topic would result in a sufficiently similar puzzle. For example, a letter-based sudoku puzzle isn't a word puzzle, since the letters (regardless of whether they spell words) could be replaced with numbers without changing the difficulty of the puzzle. If a sudoku used letters but also included clues in order to place the givens, that would be both a word puzzle and a logic puzzle.

History of Use
Word puzzles, word games, and wordplay all stem from a desire to use language in a clever way. While modern word puzzles mostly have origins in the 1900s, that linguistic desire extends as far back as written languages in general.

The popularity of word puzzles as we know them today has also risen and fallen, with many people falling into the Crossword fad of the 1920s, and interesting picking up again with the advent of the Wordsearch in the 1960s-1970s and the introduction of more rigorous puzzle-solving events in the 1980s-onward.

Subtypes
Similar to Logic Puzzles, there's a very wide range of established word puzzle types, as well as new ones being invented relatively often.

Clue-based

 * Acrostics - A puzzle in which clues are answered and individual letters from those answers get mapped to disparate positions in a larger grid.
 * Crosswords - One of the original modern word puzzles, in which clues based on definitions, trivia, and popular culture are answered and the results placed in a grid where they intersect with one another.
 * Tortured Clues - A puzzle where clues are presented having been transformed via a consistent rule. When deciphered and solved, the answers must be transformed by that same rule.
 * Word Ladders - A puzzle involving the gradual transformation from one word to another via small changes.

Rearrangement

 * Dropquotes - A puzzle where a quote or series of words/phrases are written in a grid, one letter per cell, and all of the letters in a given column get alphabetized and written above the columns. The goal then becomes to reconstruct the original string.
 * Fill-In-The-Blanks - A puzzle where a piece of text has certain words removed, leaving solvers to figure out what the original words were based on contextual clues.
 * Interwoven Strings - A puzzle where two strings of text that make sense separately (often clues) get woven together into one long, nonsensical string.
 * Trigram Hell - A puzzle in which a string of text is broken into 3-letter chunks and rearranged alphabetically (until a solver restores it, that is)

Other Word Puzzles

 * Cryptograms - Essentially a substitution cipher, often where a quote, instruction, or set of connected words/phrases are encoded.
 * Flats - A puzzle type invented by the National Puzzlers' League, where a short poem or sentence is presented and placeholder words need to be replaced based on contextual information and a particular relationship between the proper words.
 * Printer's Devilry - A puzzle where a body of text is presented, but with certain words/phrases removed after being found spanning two or more words (such as 'member line' having BERLIN removed to leave 'meme').
 * Rebus - One of the oldest types of word puzzles, in which images are displayed to represent names, words, or phrases through pronunciation or overall punniness.
 * Wordsearches - A puzzle involving finding particular words/phrases hidden within a large grid of letters.