During Lent, the 40 days leading into Holy Week and Easter and a time of prayer and contemplation for many Christians, it is traditional to give up something for the duration of Lent, as a reminder of Jesus' sacrifice.
Fat Tuesday is so called because often cooking oil, lard, butter, etc., (fat) was given up for Lent. The Tuesday before Ash Wednesday would be the day when a feast with cakes, and fried foods, etc., would be eaten to use up all the fat in the house (as katiemem pointed out).
No-one was actually expected to fast for the 40 days and nights, but simply to give something up to commemorate that period.
Nowadays, many choose to give up a favorite food or activity for Lent - anything will do as long as it is meaningful and something you will miss.
Interestingly, the Sundays during this time are not counted as part of Lent - they are a kind of mini-holiday at the end of each week.
It is called Shrove Tuesday here. The tradition was to use up the foods that people gave up for Lent by making pancakes, which led to Shrove Tuesday also being called Pancake day. Less people give up things for Lent now but lots still eat pancakes on Pancake day Shrove Tuesday.