Very aromatic and fresh, these are obviously good for curry, especially South Indian recipes. Often curry leaves are ground along with coconut or added while the seasoning is made for curry, rice dishes, chutney etc.
Store this in the fridge in air tight container, or you can freeze it.
To cook, make sure you consider the spice you're pairing this with...it doesn't complement some common herbs. Do not sub dried curry leaves for fresh, as these are not as strong.
To add to a dish consider tempering it which is frying or steeping spices in hot oil to release flavour into the oil. You can then remove the leaves and simply add the flavoured oil to your dish!