The Merriam Webster online dictionary defines "motif" as:
A usually recurring salient thematic element (as in the arts); especially a dominant idea or central theme.
The dominant idea or central theme of Nathaniel Hawthorne's The Scarlet Letter is obviously "adultery." There is probably no other novel that is dominated so heavily by one theme. The scarlet letter "A" which Hester Prynne is sentenced to wear on her breast throughout the story is a continuing reminder of the motif of adultery. She not only wears the letter but embroiders it in order to make it even more conspicuous. Her daughter Pearl is contiually touching it and asking her what it means. Arthur Dimmesdale is tortured by his part in the sin of adultery. Pearl is a product of adultery. Roger Chillingworth is determined to have revenge on Dimmesdale for his instigation of the adulterous union.
To offer examples of the motifs in Hamlet and The Great Gatsby would seem to go against eNotes' policy of answering more than one question per posting. However, I can suggest that the motif of Hamlet is revenge and that of The Great Gatsby is a combination of romantic love and social ambition in the Roaring Twenties. You ought to be able to find plenty of examples of the revenge motif in Hamlet just by reading a few of his soliloquies.
The motif in Macbeth is ambition. The motif in Othello is jealousy. The motif in King Lear is the sorrows of old age. The motif in Moby Dick is whaling. Very often the motif is named in the title, e.g., Crime and Punishment, War and Peace, Les Miserables, Remembrance of Things Past, New Grub Street, "The Hound of the Baskervilles."