Many products can connect a TV to the Internet currently, but with some limitations. Connecting to the Internet does not mean it functions like your computer, but more like the Apps on an IPhone or Smartphone.
There are a variety of ways to connect a TV to the Internet. Many TVs currently come with the ability to connect to the Internet in a limited fashion. Additionally you can use a Blu Ray player to connect a TV to the Internet, Many gaming systems can connect a TV to the Internet, and products like Apple TV or a ROKU box can accomplish this as well.
If you have or purchase an "Internet Ready TV," then what you have is a TV with the ability to connect to the Internet either wirelessly or through a hard wire to a router. Connecting directly to the router provides the best connection and download speed, eliminating much of the buffering that happens when you watch HD movies or shows. In this scenario you would run an ethernet cable directly from the back of the TV to your router.
The other option is to connect wirelessly. This is done by purchasing a wireless adapter referred to in the industry as a dongle... yes they will know what you are talking about. This dongle plugs into the USB slot on the TV and allows wireless connection. Dongles cost about $80 and are manufacturer specific.
Blu Ray players have many of the same features as the Internet Ready TVs, but also have some advantages. Almost all Blu Ray players at this point have the ability to connect to the Internet and pass this signal along to the TV, but many of them have the wireless function built in, eliminating the need for the wireless dongle. The obvious advantage is no longer needing to buy an $80 part. By purchasing a Blu Ray player to connect to the Internet versus the TV you also remove any limits you might have on TV choice, as Internet connectable TV's are less common, especially in sizes 40" and smaller.
Blu Ray players also have the obvious advantage of being Blu Ray players, meaning that with most HDTV's now your salesperson will recommend a Blu Ray player anyway, so kill two birds with one stone. As for the folks who are trying to add Internet to an analog TV, this is fine as well. Blu Ray players will still output analog signal, you just lose the benefit of better picture quality. As a last note Blu Ray players, many people are hesitant because they have large collections of standard DVD's, which is fine. Blu Ray platers play regular DVD's as well as CDs, so you don't have to replace your collection to add one.
Apple TV can be a good solution because it adds the Internet function as well as access to ITunes so you can steam music as well, but be warned it is limited on outputs. Meaning it only outputs HDMI, so it is not compatible with older TVs and it also only outputs audio in the for of an optical cable. So if you have only have the basic yellow, red, and white connections it will not work as an option for you. The ROKU box is only designed to stream Internet content to a TV, and offers only that.
There is also a product called Google TV, which is available in 3 forms. Built into a Sony TV, built into a Sony Blu Ray player, and the Revue by Logitech. Google TV offers a much wider range of Internet options, but there are some limits. It is best to go to a retailer for a demo of this product.