mode: a particular type or form of something (Random House Dictionary)
fairy tale: a story, usually for children, about elves, hobgoblins, dragons, fairies, or other magical creatures. (Random House)
When speaking of literature, a mode is a type or a form of something. For example, Pride and Prejudice by Jane Austen is a type, a form, or a mode of romance story: it is romance as a comedy of manners.
A fairy tale is particular genre of folk tale story that is usually written for children and that contains magical or mythical creatures, like the Centaurs, the talking Satyr and the talking, cooking, housekeeping Beavers in The Lion, the Witch and the Wardrobe of the Narnia series.
When asking about the fairytale mode as it is expressed in Dicken's Great Expectations, you are in fact asking: "How is Great Expectations, or a part or parts of it, similar to, reminiscent of, or like a fairy tale?" In other words, how are parts or a part of Great Expectations a type or a form (a mode) of a fairy tale?
Quite obviously, Great Expectations, with its sometimes unpleasant realism and striking social satire, is in no way similar to a fairy tale. Dickens would have been very surprised--and perhaps dismayed--to find that his novel and "fairy tale" were referred to in the same sentence. Yet there is a particular rationale behind this question.
Russian Formalism literary theory and Narratology both talk about the deep story and the surface plot of a work of literature. The deep story is that universal storyline that is found in works from every age and in many if not all cultures. Examples of deep stories might be love that conquers all; the love of a child for a pet; the enchanted magical regions of a child's world; the grief of loss. The surface story is the plot, setting, characters and dialogue that give the deep story life through any one given author.
Your question is asking you to analyze the deep story of Great Expectations and find elements of a universal fairy tale mode within it. Some possibilities for fairy tale features for this might be as follows:
hobgoblins: Magwitch might be likened to a fairy tale hobgoblin, especially as he presents himself to (or forces himself upon) Pip in the marshes.
witch: Miss Havisham can be likened to a magical, evil witch who has Pip and Estelle under her evil spell.
betrayed princess: On the other hand, Miss Havisham might also be likened to a betrayed princess who gives in to the dark side after the betrayal.
magical life: Pip's renovated life (mistakenly attributed to Miss Havisham) might be likened to a magical rescue and renewal of life rewarding the virtue of his deserving qualities like in the fairy tale about the sister who was rewarded with gems falling from her lips every time she spoke kindly or like the stories of Cinderella and Sleeping Beauty.