Two of the most popular I'm aware of in the US are Netflix and Blockbuster. Both providers give a choice of having rentals delivered for online viewing or by mail (as DVDs, which include prepaid return envolopes). As far as I'm aware of, there is no limit to how many shows or movies you can watch in a month for the fixed subscription price on Netflix. If you opt to have them delivered by DVD, you need to return one or more DVDs before they will send out the next.
Netflix's online service streams the movies to the PC, which means you can start watching the title without waiting for it to download first. The catch however is that it requires a reliable quick broadband connection. They claim to have an online library of around 12,000 movies and shows available to stream online.
Blockbuster's online service works by individual download, where you can download the titles up to 30 days before watching them. The advantage here is that a break in the broadband connection does not interrupt playback. The rental prices vary depending on the movies or TV shows you choose and expire 24 hours after the first viewing. If you prefer to watch the movies on your TV, they also have the option where you can purchase a set-top player for $99, which comes bundled with 25 rentals of your choice.
Apple's iTunes movies service works in pretty much the same way as Blockbuster's download service.
Unfortunately, due to tight restrictions by the movie industry, as far as I'm aware of, every legitimate movie download service offering major releases to permanently keep have strict limitations on how you use movies purchased to keep. For example, they may not be written to DVD and some providers (including Blockbuster) don't allow them to be transferred to another PC. If you are looking to purchase movies online to keep permanently and watch around (e.g. living room TV, laptop, at a friend's house, etc.), I would recommend sticking with the trusty DVDs.