They have fantastic movement and manoeuvrability in the air and can fly up to 30mph. They spend their early life in water then as they mature and start to fly live the next part of their life in the air. Part of their lesson can be to bring clarity (air) to something emotionally (water) complex. Likewise they can also help bring balance to situations that feel very rational and need some emotion.
Dragonflies are connected with light. Their many colours are actually often caused by how their body and skin refracts and reflects light rather than the actual pigments in their skin. They have been connected with mysticism and magic. In Japan they are often associated with light and joy.
In summary, you may wish to have near you the medicine of dragonfly if you are seeking balance between your emotions and thoughts and if you wish to be reminded of the eternal existence of all things and that all that exists (including us) is light.