A puzzle requiring a set of jumbled items (most often words) to be partitioned into distinct groups so that the members of each group share some hidden connection, which is to be identified.
A connect-wall puzzle requires the solver to partition a set of jumbled items (most often words) into distinct groups so that the members of each group share some hidden connection, which is to be identified.
The origins of this puzzle type lie in the connecting wall round from the BBC game show 'Only Connect' (first aired 2008). A similar mechanism is employed in the NYT's Connections puzzle (first published 2023).
Most commonly, as in the Only Connect original and the NYT Connections version, a connecting wall comprises 16 words presented as a 4x4 grid, and the solver is required to partition these words into 4 groups of 4 clues based on a shared connection. This is not necessarily an easy task, however, as the connecting categories are unknown and must be deduced by the solver, and the starting word set will often contain 'red herrings' - clues which might fit into more than one category. These can be disambiguated using the fact that - in a well-formed connecting wall - there is always only one way to partition the set correctly.
Guessing groupings at random is a poor strategy, as there are $\dfrac{16!}{4!^4}=63,063,000$ possible partitions per grid. Instead, solvers must usually rely on spotting hidden wordplay and using general knowledge to identify the connections.
As the genre has been explored further within this site, certain PSE-specific quirks have appeared over time:
connect-wall puzzles on PSE often employ an additional 'final step' in which the four group connections are themselves connected in some way - identifying this meta-connection is often the end goal of the puzzle. Good examples by different PSE users can be seen here, here and here.
Whilst the original Only Connect connecting wall only uses words as its constituent items, here on PSE anything is fair game! Within this site's archive can be found connecting walls based on images, cryptic clues, movie trivia, sheet music and much else. Many other examples of creative connecting walls can be found in the November 2020 Fortnightly Topic Challenge on 'Wordless Connecting Walls'.
There's nothing to say that a connecting wall must comprise only 16 items or that each individual connection category must have exactly 4 members, that each connection category must have an equal number of members (as long as the intended group sizes are specified elsehow, e.g. GCHQ Where? style puzzles), or that the connecting 'wall' must even be a rectangular two-dimensional wall (how about a Fano plane or a three-dimensional cube of walls?!). If the question involves partitioning a number of items into a number of categories having a number of members, it's a connecting wall!
There is a generally-accepted convention within this community that - unless otherwise stipulated by the OP - partial answers suggesting some but not all category connections (i.e. without fully resolving the wall) are discouraged. Since connecting walls often contain red herrings (items that might fit into more than one group), the complexity (and beauty) of this puzzle type lies in being able to deduce a complete partition of the starting item set, rather than just being able to find a link between a smaller number of words.
Those interested in creating their own (harder) connect-wall puzzles may find it useful to read this question and its accepted answer, which offers advice on how to do that.
