Unlike dry spices, not many fresh herbs are used in Indian cuisine.
Cilantro (Hara Dhania)
Cilantro, the leaves and stem of the Coriandrum sativum plant, is used to garnish almost all Indian dishes. You probably already know what it is. You can grow cilantro yourself; it grows back if you cut it and keep the root in the soil.
There are many kinds of mint, but the most common is spearmint, the Mentha spicata plant. Mint, along with cilantro, is part of green chutney, the sauce that goes with many Indian street foods. You can grow mint yourself.
Curry Leaves (Kadipatta)
The leaves of the Murraya koenigii tree are used in almost all South Indian foods. They are almost impossible to find in the USA (just once in a while they pop up in an Indian grocery store), and they go bad in only a few days in the refrigerator. If frozen or dried, they lose flavor. If you are interested in South Indian food, I recommend you to buy a Murraya koenigii tree, which many Indian people own.You can also just buy a small amount at a time if you do not use it every day. If you cannot find it, you may skip it from a recipe. It will lack the curry leaf flavor, but it will still be delicious. Obviously do not do this in a recipe for curry leaf chutney; in that case just make coriander or mint chutney.
These leaves are not related to curry powder.