Imply is a needlessly ambiguous word, in that it currently has two common and easily confused meanings. The most common is to mean 'to suggest'. This is particularly used in informal speech, and has an overtone of being more emphatic than simply saying 'suggest'.
More formally, in the realm of legalese and logic, imply means 'to require', although it will be more often defined as 'logically entail' or 'legally entail'. It is not common to use this sense in conversation, but it does appear in certain popular phrases -- such as 'correlation does not imply causation', or 'implied consent'.
As you can see, the statements 'correlation does not logically imply causation' and 'correlation does not suggest causation' are very different -- most relevantly in that correlation does often suggest causation. Likewise, 'you have legally already given consent' and 'you have suggested that you may give consent' are entirely different beasts.
Matters are confused a bit further by the fact that many pundits and politicians who use the word in the less formal sense would quite like for people to interpret it in the strong sense. While they can't get away with the claim that their arguments are airtight, they are more than happy to help people overlook weaknesses in their thinking. It is often impossible to say if this sort of evasion is intentional or not, but it is omnipresent in the American political scene.
My personal preference is to declare the more technical and precise definition the valid interpretation, and the common usage a vile abasement of the English language. Of course, the much more practical response is to forgo the usage of imply altogether, aside from those situations where you are clearly referring to logic or law. There is no need for most of us to use the word imply in conversation, as the words suggest and entail already do exactly the same job, without any confusion.
A word bank for writers:
Suggest: indicate, intimate, allude, refer to, connect to, strengthen, suppose, reinforce, connote, correlate.
Entail: require, mandate, demand, necessitate, follow.