Flower vases can be especially tough to clean. This is usually because the stagnant water contains trace amounts of lime and other mineral deposits which cling to glass and resist attempts at cleaning with mild cleaners. If you have a vase that just won't come clean, use this process and dissolve those deposits away.
Empty your stained vase, and rinse it with cold water.
Pour vinegar into the vase until you've filled it about halfway up to the stain line. Then add water until you've covered the stain line.
Drop in some tea bags. Smaller vases will need about three bags, while larger ones may need six or more. There's no exact science to how many you should use. Just imagine that you're making a very strong batch of tea, and adjust for the volume of liquid that is inside the vase.
Set the vase aside for at least a day. If you can wait two days, that's even better.
Pour out the mixture, and dispose of the tea bags. Immediately rinse the vase with hot water, then scrub the inside using hot water, liquid dish soap and a bottle brush. The mineral deposits should be weak enough that they will either simply rinse away or can be easily scrubbed off.
If your vases develop deposits very quickly, you may need to improve the water softening and filtration systems in your home. Using distilled water will help you avoid these deposits from the outset.
Do not cover the top of the vase after filling it with the vinegar mixture.