The first word I'll look at is immoral, which means 'not consistent with moral law or standards' or 'ethically wrong'. The word dates from the mid 1600s and is combination of 'in', meaning not, and 'moral'. Interestingly, when used as a legal term, immoral simply means 'contrary to common good or reasonable order'. Someone who is immoral, then, is not following established laws and standards. This could be anything from being involved in an affair to breaking laws because they aren't convenient. An example of using immoral is:
Even though Joss liked John as a person, as a police officer she couldn't help but find his wanton disregard of the law extraordinarily immoral at times.
Amoral, on the other hand, means 'lacking a moral sense' or being 'unconcerned with the rightness or wrongness of something'. So a person who is amoral might not be able to tell whether something that they have done is right or wrong, or simply may not care. Again being a combination, of 'a-' meaning not or without and 'moral', amoral is a far more recent word. First used in the 1880s, and possibly created by Robert Louis Stephenson simply as a differentiation from immoral, an example of using amoral is:
While Harold held himself to a very strict code, he considered John's approach more amoral -- he would do what was needed to protect the client.
As you can see from my examples, which both refer to the same character, not only is there a difference the meaning of amoral and immoral, but there is a difference in tone as well. Calling someone immoral is a judgement on their behaviour, but the word amoral is more impartial or neutral. Someone who is immoral violates norms, while someone amoral has no understanding of them. So when using these words, think about both the meaning and also the tone or association that you want to use. After all, if someone thinks your usage of amoral is immoral, they certainly aren't thinking very highly of you.
Amoral definition: https://en.oxforddictionaries.com/definition/us/amoral, http://www.etymonline.com/index.php?term=amoral
Immoral definition: https://en.oxforddictionaries.com/definition/us/immoral, http://www.etymonline.com/index.php?term=immoral