Note: This article is very U.S.-centric; sorry to the non-Americans!
1995-2000: The online-banking stone age
It's worth briefly mentioning the rise of online banking in the 1990s as background. As the internet began to become popular amongst the general public, banks began offering online access to their accounts. This led some entrepreneurs to think: Many people are doing much of their banking through a combination of ATMs, checkwriting, and the internet, and you don't need branches for any of that. So, a slew of online-only banks was founded amidst the dot-com euphoria (many, as you might guess, no longer exist; anyone remember CompuBank?). These banks typically offered a similar range of products to traditional banks, and indeed tried to play up their similarity with "real" banks. To hook consumers, they typically offered lower fees and higher interest rates, but only a limited number of consumers were interested.
Fall 2000: ING Direct
In the fall of 2000, Dutch banking giant ING launched a different, and ultimately more successful, take on online-only banking with ING Direct. Rather than an online analog to a brick-and-mortar bank (ING was already a brick-and-mortar bank), ING Direct was (and is) a simple, no-frills savings account paying a high rate of interest, but with no checks, no ATM cards, and basically no other services besides an account in which to park money. Customers are expected to have a local checking account they use for their normal banking, and then transfer money between that account and ING Direct via ACH (automated clearing house, a system of electronic bank-to-bank transfers that typically takes 2-5 days). Somewhat notably, they didn't skimp on customer service--they reportedly have had good email and phone customer service, and now in some parts of the country they have a strange network of brick-and-mortar locations that provide in-person customer service but aren't bank branches, the ING Direct Cafés.
2003-present: More competition
ING Direct was without direct competition for its first three years, but now there are quite a few others offering essentially the same account, including some online-only banks (who arguably should've been the first ones on this bandwagon). I'll mention only a few of the more notable ones.
The next major entrant after ING was in fact an online-only bank, VirtualBank. It had been founded in the late 1990s as the online analog to high-class private banking, aimed at newly-wealthy techies. In mid-2003, they repositioned themselves and offered an account virtually identical to ING Direct's, which they called eMoneyMarket, and which followed the same strategy of no frills and high interest rates (the "money market" here means a money-market deposit account, which is FDIC-insured and basically a savings account; not to be confused with money-market funds, which are a different beast). Being a new entrant, they typically paid a few tenths of a percentage point higher interest than ING Direct, and so gained some customers.
The market expanded rapidly through 2003 and 2004, and a bunch of others began to open similar accounts by 2005. Emigrant Direct, the online initiative of a one-branch local New York City bank (so basically an online-only bank), caused a bit of a stir in January 2005 when they launched a 3.00% APY account at a time when the next-highest rates were 2.25%. HSBC, a global banking giant, entered the online-savings market with HSBC Direct in March 2005. This is an interesting variation, because it adds back in one frill: ATM cards that can be used at HSBC's brick-and-mortar locations worldwide, with very low currency-exchange costs--an interesting attempt to turn their large global network of branches into an asset even in the online-banking world.
Links to more information
BankRate.com seems to have become the canonical site to look up interest rates on. It doesn't have a specific section for online accounts, but you can search for rates on "MMAs" (money-market accounts) in the "checking & savings" section. It's useful to get an idea of what's out there, but beware of simply picking the highest rate that comes back in your search and signing up.
The Fatwallet.com forums are one of the best sources of information, and include some very long threads discussing many of these accounts. As with any forum it takes a huge amount of digging and reading to get the good information, but for raw information it's hard to beat. A good place to start is the Best Current APRs thread, which also links to a bunch of individual threads on specific banks.
I maintain a fairly bare-bones site at pfstuff.com, summarizing the interest rates and pros/cons of accounts that offer good rates but don't have any fees, minimums, or other strange requirements. I've also collected some information on the sign-up bonuses some banks offer, and pieced together enough information to plot historical graphs of various banks' interest rates.