Think of a successful boxer at a stool with multiple legs. The legs are Chin, Power, Technique, Footwork, and Strategy/Mental Discipline. If just one of these legs are lacking the stool is more likely to fall over, but hopefully the boxer is facing a chump who lacking even more. If several of the legs are bad the stool is much more likely to fall over. Thus a good boxer should have a good balance in all 5 key areas in order to be successful.
Quality footwork allows a boxer to stay balanced and on their feet, and also gives them an advantage because they can move in closer faster and weave around / fade from punches more easily. This gives them both an offensive and defensive advantage.
A boxer with poor footwork will get hit more often, knocked down more easily, and if their chin is poor also (or suffering due to being given a beating) will be less likely to get up within 10 seconds and lose a match due to a technical knockout. (And being KOd many times looks bad on your record if you want to get into amateur or professional boxing competitions.)
Examples of great boxers who were really good at footwork: Manny Pacquiao, Joe Calzaghe, Oscar De La Hoya. Watch some of their videos on YouTube and you will see some amazing examples of footwork. Watch it on a big screen so you can see better and pay close attention to how they move their feet.
#1. Use footwork to get out of range (fading).
To get good at this practice running backwards. Try to get really good at running backwards.
#2. Use footwork to weave sideways.
A good way to practice footwork is to position yourself in a room with your left side against a wall. Move sideways to your right by moving your left foot forward and to the right, and then your right foot behind and to the right. Then your left foot switches and goes behind and to the right, followed by your right foot forward and to the right. Continue alternating which foot is forward and which is behind until you reach the opposite wall. Then switch directions and go left, each time alternating which foot is in front and which is behind. NOTE: Make sure your shoes are tied properly and fit okay. This is an easy exercise to mess up and you can trip yourself if you get confused. The exercise is often used by football players as sideways footwork is also important for football.
#3. Step forward into punches.
Regardless of whether you are doing a straight punch, a hook or a cross you want to step in with your punch to add your body weight and additional momentum with the punch. A way to train for this is jogging or sprinting, while using your arm motion during jogging so you look a bit like rock-em-sock-em robots (if you remember that old 80s toy). I admit that is a weird analogy, but it works as you should be able to visualize it.
#4. Get better balance by doing Balance Exercises
There are a number of ways to build better balance using footwork, but the one I like to recommend to beginners is to practice hopscotch. (You may feel silly doing this, but it really works.) The trick is to challenge yourself physically so you can do hopscotch routines quickly on either one or both feet.
Another good balancing exercise for your feet is rope skipping. Learning to skip rope both fast and stay coordinated will build your leg muscles, coordination and speed.
#5. Keep exploring more footwork exercises.
I have only covered the basics here. The examples above will help a boxer gain muscle memory in their legs, more balance, more speed, more coordination and give them an added edge in the boxing ring. A nimble boxer who is well rounded and can pack a punch - or float like a butterfly, sting like a bee (Muhammad Ali quote).
I will go into more detail about some of these footwork drills on later dates but for now I recommend doing the exercises listed above.
There are a variety of other exercise drills out there aimed at boxers who want to improve their footwork. In the video below is an example of an exercise designed more for kickboxers and muay thai boxers, but is beneficial because it trains the abdomen, shoulders and legs - and improves balance