Like Fran, I'm familiar with Aldi from Germany, and I often shop at one in the U.S. when I'm visiting my family. Over here, Aldi has customers from all income classes, and the store brands are generally branded goods packaged especially for Aldi, and much cheaper than at "traditional" supermarkets. The same is true of Lidl, another big discounter, which I recently read is expanding to the U.S.. The quality of the produce seems to depend on the locality.
Aldi is typically German in the "coin-operated" shopping carts and "Bring Your Own Bags"; as Jenny pointed out, the latter is usual in Europe. I normally have a cloth shopping bag tucked into my handbag or backpack, and a lot of people just keep a box or washbasket in their cars for grocery shopping.
If you don't mind not having a big variety of brands to choose from, Aldi can really save you a lot of money.