To imply is to express something indirectly. For example, you might imply that it’s time for a guest to leave by saying that you are getting tired. To infer is to surmise or conclude, especially from indirect evidence. For example, if you were to tell a guest that you’re getting tired, the guest might infer that it’s time to leave.
More broadly, infer means to deduce. For example, when the sky grows dark in the middle of the day, you might infer that it’s probably going to storm.
Infer has been used in place of imply so often and for so long that some dictionaries now list it as a synonym of imply in a secondary sense. In edited writing, however, the traditional distinction is usually kept intact.
Her comments imply in a sly way that there is possible malfeasance by this administration. [Knoxville News Sentinel letter to the editor]
He seemed to imply that many of the debts in question are for items people failed to return after the bankruptcy was announced. [NASDAQ]
So all scientists can tell about each boat is where it is and how fast it moves, and they have to infer when it was actually fishing. [Chem Info]
On the table sits an untouched breakfast—the sodden castoffs, we infer, of the insolent child. [Time]