I am a big believer in teaching martial arts from the ground up.  This literally starts with understanding your feet and the floor you are stood upon.  My best advice for anyone who wants to improve their sparring or the application of their martial arts skill is simply to work on developing their stance and footwork.  With that in mind, here are my top 10 tips for improving your footwork
1. Get comfortable with both stances (it helps create balance and a wider skill base if you regular switch from orthodox to southpaw in training)
2. Commit to constantly checking (or even better, get your coach to check) your stance is correct until it becomes second nature (a large number of boxers regularly train with a piece of equipment that ensures the stance remains the same whilst moving around.  There is a product called fast feet that is available within the UK from stockists such as www.sugarrays.co.uk, which is a good representation of this)
3. Skip (skipping develops the calf muscles and is great for developing not only fitness and endurance but also timing and rapid foot movement) regularly
4. Start simple.  Don't overcomplicate your footwork to begin with.  Work and drill in isolation moving forward, back, left and right.  Then drill rounds working this movement in combination.  Once you start to feel comfortable and have balance moving in any direction, start to add techniques singly onto the beginning or end of the footwork
5. Understand your distancing.  For a drill to help you with this, you will need some coloured tape and a bag.  Put lines of tape on the floor that mark the distance away from the bag that mark where (a) you can reach with a straight punch, (b) you can reach with a kick and (c) you cannot reach with any technique.  These won't be that far apart.  These small movements and their practise will help greatly with the ratio of the techniques that are thrown/landed during training and sparring as you start to intuitively grasp where you need to be in order to hit
6. Introduce over types and variations of footwork one at a time in isolation before adding into the mix combining pieces together in drills prior to constant practise via shadow boxing and training.  An example of this would be to introduce the side step.  Practise in both directions in both stances prior to combining with footwork already practised ie move back one step, side step to the left.  Make sure you work progressively with this.  See my lesson plans on improving footwork for an example of a laid out structure
7. Use a partner.  A great drill to utilise is to mirror a partners footwork.  One person should lead the drill (for a 2min round prior to swapping roles) by moving one step using any of the practised pieces of footwork.  The other person should then mirror that footwork to try and stay in exactly the same place/distance from their partner.  This should be started slowly but then as confidence improves it can be speeded up and made more difficult (by utilising combinations of footwork).  
8. Integrate your footwork into every aspect of your training.  Create drills with your coach that allow you to practise each movement on varied pieces of equipment.  A good starting point would be executing single techniques utilising the correct distance moving forward, back, left and right on focus pads.
9. Create sparring drills that allow you to practise your newly found footwork within its correct context.  For example, one person attacks with just the jab (lead hand straight punch) whilst the other tries to defend primarily utilising footwork.  
10. Free spar.  Lower the contact so that you are not overly preoccupied with being hit and work rounds with as many different bodies as you can.  Constant practise of your footwork in this environment will ensure you will see the results of all your hard training as you become both more elusive at avoiding being hit and more effective at landing the shots you throw

I have used the kind of footwork that is commonly found in styles such as boxing, kickboxing and thai boxing but the principles would remain correct for any other discipline as well (with a little tweaking on the specifics).  
Happy training and to quote a man well know for his speedy movement...
"Float like a butterfly, sting like a bee. His hands can't hit what his eyes can't see. Now you see me, now you don't. George thinks he will, but I know he won't" - Muhammed Ali