The little packets that come with some cut flowers contain biocide to kill bacteria, an acidifier to adjust the water's pH (which helps the movement water and food), and a sugar to nourish the flowers. Instead of using that stuff, some basic knowledge and attention can help keep your flowers happy.
1. Cut the stems at a 45 degree angle and don't let leaves sit in water. The angle increases absorption, and soaking leaves will decay.
2. Change the water and cut the ends daily.
3. Keep your flowers away from direct sunlight and ripening fruit. The latter emits ethylene gas; this hormone, an expert from the Brooklyn Botanical Garden explains, "causes buds to remain closed, petals to have poor color, and flowers to have a shortened vase life."
4. Add a little sugar or even a penny. The sugar will feed the flowers, but it can cause bacteria to grow (so it's even more important to change the water regularly). The copper in pennies is thought to work like an acidifier.
To see a comparison of how tulips treated with fresh water, sugar, bleach, pennies, Listerine, aspirin, and plant food match up over a week, see this slide show from Real Simple. For more tips on care for specific types of flowers, follow advice from the Brooklyn Botanical Garden.