A cryptic crossword is one where the clue has two parts: definition and wordplay.
An example is, "Spanish is, confused parent circle concludes, a made up language (9)". The answer is "Esperanto".
"A made-up language" is the definition - it defines Esperanto. But there are lots of made-up languages, so the precise answer has to be found using the wordplay.
The Spanish word for "is" is "es".
"Confused" is a way of saying "all muddled about", and if you middle about the letters in "parent" you get "perant".
"O" is a circle, and it concludes the word.
For a crossword to be cryptic, it should have clues of that nature.