The Difference Between Fasting and Abstinence:
Fasting and abstinence are closely related, but there are some differences in these spiritual practices. In general, fasting refers to restrictions on the quantity of the food we eat and on when we consume it, while abstinence refers to the avoidance of particular foods. The most common form of abstinence is the avoidance of meat, a spiritual practice that goes back to the earliest days of the Church.
Depriving Ourselves of Something Good:
Before Vatican II, Catholics were required to abstain from meat every Friday, as a form of penance in honor of the death of Jesus Christ on the Cross on Good Friday. Since Catholics are normally allowed to eat meat, this prohibition is very different from the dietary laws of the Old Testament or of other religions (such as Islam) today.
In the Acts of the Apostles (Acts 10:9-16), St. Peter has a vision in which God reveals that Christians can eat any food. So, when we abstain, it's not because the food is impure; we're voluntarily giving up something good, for our spiritual benefit.
Current Church Law Regarding Abstinence:
That's why, under current Church law, the days of abstinence fall during Lent, the season of spiritual preparation for Easter. On Ash Wednesday and all of the Fridays of Lent, Catholics over the age of 14 are required to abstain from meat and from foods made with meat.
Many Catholics don't realize that Church still recommends abstinence on all Fridays of the year, not just during Lent. In fact, if we don't abstain from meat on non-Lenten Fridays, we're required to substitute some other form of penance.
For more details on the current Church law concerning fasting and abstinence, see What Are the Rules for Fasting and Abstinence in the Catholic Church?
Observing Friday Abstinence Throughout the Year:
One of the most frequent hurdles encountered by Catholics who abstain from meat on every Friday of the year is a limited repertoire of meatless recipes. While vegetarianism has become more widespread in recent decades, those who eat meat may still have some trouble finding meatless recipes that they like, and end up falling back on those staples of meatless Fridays in the 1950's—macaroni and cheese, tuna noodle casserole, and fish sticks.
But you can take advantage of the fact that the cuisines of traditionally Catholic countries have an almost limitless variety of meatless dishes, reflecting the times when Catholics abstained from meat throughout both Lent and Advent (not only on Ash Wednesday and Fridays). You can find a nice selection of such recipes in Lenten Recipes: Meatless Recipes for Lent and Throughout the Year.
Going Beyond What's Required:
If you would like to make abstinence a bigger part of your spiritual discipline, a good place to start is to abstain from meat on all Fridays of the year. During Lent, you might consider following the traditional rules for Lenten abstinence, which include eating meat at only one meal per day (in addition to strict abstinence on Ash Wednesday and Fridays).
Unlike fasting, abstinence is less likely to be harmful if taken to extremes, but, if you want to extend your discipline beyond what the Church currently prescribes (or beyond what it has prescribed in the past), you should consult your priest.