A logogriph is a word puzzle, which I'm afraid will probably involve anagrams. In its grandest form the author will cut the one-word answer to the puzzle down into its various constituent parts and give clues to each part, and then, at the end, a clue to the whole thing.
Yes, they do tend to be quite long-winded, and in this age of the one-liner they have fallen, thank heavens, out of favour. Formerly, though, people actually found them entertaining and even respectable people made them up.
Here's one by Lord Macauley:
...and for those of you who have now lost the will to live, which will surely be most of us, the answer is below.
A logogriph can also be a puzzle consisting of two words which give clues to two other words which are anagrams of each other. For instance QUIET HEARING gives you SILENT LISTEN.
On the whole, I wish the logogriph had been a fabulous beast from Alice in Wonderland. Well, it would have been more fun, wouldn't it?
Word To Use Today: logogriph. This word is made of the Greek logos, word, and griphos, puzzle or fishing basket (presumably one like a lobster pot, from which the prey can't escape).
The answer to Lord Macauley's logogriph is the kind of fish called a cod. Head cut off: od (which sounds like odd, geddit?); tail cut off co, as in company; middle consists of o, or zero; head alone c, or something which sounds like sea; tail alone d, which sounds like Dee, a British river.