A recommendation engine tries to predict how much a user will enjoy certain goods (movies, books, songs, etc) and makes recommendations. They are often used by online vendors to suggest new purchases.
A recommendation system, or recommendation engine, is an automatic tool that recommends goods to a specific user. There are many examples on online commercial websites: Netflix or IMDB recommends movies, Amazon recommends books any many more, etc. The input of such algorithms are most commonly the past purchases of the user, the rating the user gave to other goods, the items that have been purchased together ("customer who bought this also bought..."), the browsing history, etc.
See also wikipedia.
Reference: adapted from Stack Overflow SE.