Although often confused with the word "imply," the word "infer" has a totally different meaning. When you infer something, you draw a conclusion based upon suggested information, as compared to concrete information. For example, if your mother offers you money without you having asked for it, you can infer that she assumes you are going broke; however, if you examine your bank account and realize you have no money, you're not inferring your lack of funds, but drawing a conclusion about your funds based upon concrete evidence. In other words, to infer simply means to assume without factual evidence.
When you imply something you suggest a fact, opinion or truth without stating it directly. For example, if you told someone they should wear a jacket and tie to dinner, you are implying that the gathering is going to be formal. Once you make an implication, the other person can infer a conclusion. Using the jacket-tie example, the other person would either infer that it's a formal gathering or that they are not currently dressed properly for dinner. You can only infer something after an implication has been made.
The word "imply" has its roots in Latin according to Educationbug.org. The original word -- implicare -- means to entangle or unite. The word "infer" also has Latin roots. The original spelling was inferre, which translates to bring and bear. Both words are considered relational- and/or converse-pair antonyms. This means that infer and imply are converse opposites of each other. Other examples of converse opposites are husband in relation to wife, and buy in relation to sell. This is different than simply being opposites of each other, like black and white.
You can imply something with a specific tone of voice or with a specific look. Silence can also be used to imply something. For example, if you ask your teacher about your exam grade and she says, "You did really well," with a sarcastic tone -- or says nothing at all -- you can infer that you failed that exam, or at least didn't do as well as you had hoped. If you ask your teacher if you got an A and she raises an eyebrow, her look can imply that you did not.