On Selenium download page, there is a link to the Selenium RC (Remote Control).
On another Selenium download page, there are links for a Selenium RC 1.0.3
You should take a look at the documentation outlining the different parts of Selenium. That's a good place to start as it guides you through the process of getting setup. Full disclosure, I help write the docs.
As for those downloads, the first link you have is the official releases for the project. The second link is the current beta builds, once they hit 2.0 they'll be on the site.
Then we get to the Selenium 2 beta. Selenium 2 is a major departure from the Selenium 1 model because it doesn't require a Selenium server. I say 'require' because it's optional to run the tests remotely on another computer. Selenium Server Standalone is the server you'd use for this. It's compatible with Selenium-RC as well as Selenium 2 for remote purposes.
You may have seen Selenium 2 referred to as WebDriver. WebDriver was another project that was merged a couple years ago and became the basis for Selenium 2. That's why Selenium 2 has a WebDriver interface, sometimes called the "WebDriver" api to distinguish from Selenium-RC.
If you're just starting out, I'd take a look at Selenium 2. It's getting 99.9% of the developer love right now, and the Selenium 1.x apis won't be advancing any further. As of January 2011 the Java libraries are the best supported, followed closely by .Net and Python/Ruby. Watir (the popular Ruby browser automation library) uses selenium under the hood if you want another api option.
The documentation on Selenium's website says that the Remote Control consist of
The server is responsible for doing the actual work (executing actions in the browser) while your specifications for actions (the control of execution) comes from the different libraries.
This is called the Remote Control because the server does not have to be deployed on the same machine where the actual scripts are executed. The scripts can connect to a remote selenium server over the network and execute actions.
To sum up, when you are using the Selenium Remote Control, you are using the server + client libraries. They are meant to be used together.
I do not know if this is going to be the same for Selenium 2 or if they are changing things. I am only confident this is how things work for Selenium 1.