Once you have grilled the eggplant, it will keep in your fridge for 4-5 days. It can be eaten cold or you can reheat it in a hot skillet or in the microwave.
If you haven't cooked the eggplant yet, they generally last for 2-3 weeks when kept whole in the fridge. On the counter, they will last 4-5 days. It is generally not recommended to freeze raw eggplant, but you can freeze cooked eggplant.