The term "baptism of fire" essentially refers to a person's initial experience when trying something he has never previously attempted. In addition, the new experience is usually difficult and the results will either make or break the person. Baptism of fire is an expression that originally derived from the book of Matthew in the Bible, but later became a saying that relates to a soldier's first time engaging in battle. In the modern context, the term can relate to a variety of circumstances in which an individual is placed into a difficult situation for the first time and tested.
In Matthew 3:11, John the Baptist used the term "baptism of fire" as a way to encourage sinners to repent. Basically, people who repented would be baptized in the Holy Spirit. Those who did not repent would baptized in fire. While this context is the first recorded use of the term, it's meaning has changed slightly over time.
In addition to the biblical context, "baptism of fire" was used by Dr. Barry Edward O'Meara in a memoir he wrote about Napoleon Bonaparte. The term that O'Meara used was in a military sense and related to a soldier who was seeing battle for the first time. Basically, a soldier's first time in combat was a difficult experience that made or broke him, and was therefore dubbed a "baptism of fire".
In more modern times, this term has come to mean any type of difficult experience that a person engages in for the first time. Usually, this means that an individual is out of his comfort zone and exposed to a situation that he is unaccustomed to. During this time he must quickly learn how to cope and thrive from the experience, or ultimately fail.
One example could be a new employee starting a job with a very demanding customer service position. His first day on the job would be a baptism of fire, in which he faces things like learning new systems, dealing with irate customers, managing stress and other difficulties. At the end of the day he will either do well and thrive, or fail and is unlikely to keep the position for long.
Another example could be a babysitter on her first sitting assignment. She may be faced with a few unruly children with tempers or other problematic issues. During this time, her patience will be tried and her skills will be tested. Her first day would be her baptism of fire, and she might adjust to the situation and find it rewarding. Otherwise, she might find the experience miserable and fail.