To add a bit of description on terminology to what’s been said:
The “Flush” refers to when the leaves are harvested. As far as I know, this mainly applies to Darjeeling. Tea leaves are harvested in Spring - “First Flush” is earlier in the season, and “Second Flush” is toward the end. There’s not a massive difference (though if you compare two directly and have had tea before, it’s noticeable). Second Flush has a stronger flavor, and usually commands the higher price. If you’re going to get that detailed go for the Second Flush (based on what you say you like).
Among Darjeeling teas, you can also get “Single Estate” and even from some particular estates (though I believe some of the ‘estates’ may cover several growers). This starts to get similar to Scotch (with ‘single malt’ vs. blended). Personally I think there’s greater variation between harvest year than between estates, and there’s not enough difference in either to really matter much. Most of the time you’ll have blends, which can be really excellent (again, it’s like varieties of Scotch).
As for the regional names, each has its own flavor :
Darjeeling tends to have strong flavor. It has a heady but slightly flowery aroma and a distinctive sharpness. With higher quality the aftertaste is pleasant despite the tea’s astringency (to my tastes). The majority is sold as black tea, but you can get green and even white tea from there (and you should try it just once if you can. It’s not highly vegetal like most Chinese green teas - it’s more of the Darjeeling floral quality, with less of the complexity and astringency of the black tea).
Assam tea is malty (in a slightly earthy “peaty” way) and mellow. It’s mild (but good ones are full-bodied). It’s also know for its deep amber liquor (in Assam, tea without milk is called “Red Tea”). If you’re ordering it on its own, just make sure it’s “Orthodox” [Whole Leaf]. Most Assam tea is of the cheaper, machine-processed type (CTC) intended to be boiled and mixed with spices for milk tea (roughly what’s known in the US as ‘chai’).
Ceylon tea comes from Sri Lanka (too bad we couldn’t call it Serendipi). It has a very strongly fruity, slightly sweet flavor. On its own it can smell a little like rose hips or strawberries. It often appears in blends, but can be nice straight, if a bit forceful.
Nilgiri tea is a less common variety. It comes from down south as well (Tamil Nadu). I’ve never tried it, but it’s supposedly similar to Ceylon.
There are a few other areas in India that grow tea, but I’m not too familiar with all of them.
[Right now I’m drinking a Darjeeling Muscatel (blended by this woman)].