OTA - Over the Air. This means you can download the application through your BlackBerry browser and install it straight onto your BlackBerry. You don't need the BlackBerry Desktop Manager for an 'ota download'.
App - Application. Just like you can install extra software onto your computer/laptop, you can install extra software onto your BlackBerry. A software program for your BlackBerry is called an 'app'.
RIM - Research In Motion. The company that makes the BlackBerry.
BlackBerry Desktop Manager - The CD and software program that came with your BlackBerry. It runs on your PC and allows you to do all kinds of stuff with your BlackBerry, like backup all your data, install new applications, etc.
3rd Party - This is when a company other than RIM has made the BlackBerry app. RIM makes some apps for your BlackBerry (like the facebook app, and flickr app, but most BlackBerry apps are made by other companies. Usually they will make a profit by selling their apps, but sometimes offer them for free. Sometimes they also offer the basic app for free and the upgraded 'premium' version for a fee.