Zappos has fantastic customer service, great selection of shoes, and good prices. They've been owned by Amazon since 2009.
AbeBooks is a fantastic used (or new) book marketplace. Amazon bought them in 2008.
Audible is the largest retailer of audiobooks. Amazon bought them in 2008. Try buying an audiobook anywhere else.
Amie Street provided independent music downloads will a revolutionary demand-based pricing system. Amazon bought them in 2010 to shut them down.
Woot was a pretty nifty daily deal site. They were bought by Amazon in 2010, and have really lost their original focus since.
If I want to buy just about anything online, I'm stuck buying it from Amazon. If it happens to be something that Walmart or Target stocks, I could order it from them instead. They charge $5+ for shipping even the smallest items though.