Implication: in utterances, an implication is a meaning the hearer/reader discovers by comparing the words' literal meaning with other contextual cues which add to, subtract from, or completely change its ultimate significance.
E.g., in reply to your question "will Professor X cancel class today?" your roommate, who has taken several courses with Professor X says "Fat chance!" Ordinarily, chances are not "fat," but metaphorically a "fat" chance might be said to be a likely chance as opposed to the idiom "slim chance." However, there is in your roommate's intonation a sarcastic exaggeration which warns you s/he implies "no chance!"
In New Criticism, analysis of words' implications is the process by which words' connotations are derived, and many implications may operate in a single poetic utterance, because of the complexity of poetic language. See definitions of "irony," "ambiguity," and "paradox" for examples.
"To imply" is a verb which the work can do. (E.g., "Lear tells Cordelia 'Nothing will come of nothing,' a declaration which implies that she should supply 'something' in the form of flattery or she will lose the competition for her share of the realm.") Readers can never "imply" something from the text, though they may, by writing vaguely, "imply something about the text.". "To infer" is a verb which the reader can do. Readers can "infer" something from the text, as when a text's implications lead them to infer that the author is speaking ironically. (E.g., "Lear's eldest daughters refer to him scornfully, from which we may infer that he treated them badly when they were younger.")