Your description of wordpress.com as a hosting company specializing in WordPress blogs is correct. But wordpress.org is the Web site of the developers of the software, and it's from there that you download a copy; but it doesn't host or run the software for you. Just so we're on the same wavelength: You're talking about an installation of WordPress software on your own Web site, right?

On that assumption, while there may be some special widgets and add-ons developed exclusively by wordpress.com, for the most part--by which I mean by the tens of thousands--there's a marketplace for add-ons identical in concept to an Apple or Android app store. http://wordpress.org/extend/plugins/.

I use that catalog for shopping, but when the time comes to download and try out something, I always do it from inside WordPress, using the Plugins section of the dashboard. You can search the catalog again from inside, to find your pick; and then just tell WordPress to download and install it for you. It's either smooth and seamless (most of the time) or crashes and burns (occasionally). In the latter case, you've learned the most basic fact about a plugin, it's POS quotient.

You won't need to touch a single line of code to do this. But it sometimes helps. Very few plugins are created by professionals, and the quality varies as wildly as it ever did in the days of DOS and shareware. Caveat eternium.