Mods, feel free to move to a more appropriate location, duplicate, or sticky.
Okay, since Android is new to most of us, and I see alot of people on the comments that have trouble using things they find on the Market, I figured I should explain what the difference is between an Application, Widget, and Theme (or other add-on) and how to run each of them.
These are actual programs that run on your phone. When you use them, they open and take up the whole screen. If you want to use another application, you have to multi-task away from an application. They do not run on your home screen like Widgets. (although some applications also include a Widget so you can have quick access to the application, more on this later) Some included applications on most phones are the Browser, Navigation. Other examples are the Bank of America app, PicSay, and most games.
You open the application from your drawer, icon on your home screen, or other location on the phone. Basically you click and the application runs, nothing fancy. Some people assume they all work like this, that brings us to Widgets.
Widgets on Android are much like Widgets on Windows Vista if you've used it before. They are streamlined versions of applications that are designed to run directly on your home screen (aka. workspace). They typically have at a minimum, the more important features of a similar application. You can have multiple widgets on a single home screen, and they serve as "quick access applications". Sometimes these widgets are streaming information from the app to your home screen, such as the case with Facebook and popular Sports apps, for example.
Examples of widgets would be any of the toggle buttons (wi-fi, gps, etc.), weather, Google Search bar etc.
Some Applications also come with widgets included with them. These act as a way to access/control the app right from the home screen. For example a music player application may have a widget to skip songs, pause, play, etc. right from the home screen without having the application open. The Facebook widget shows you updates from the Facebook application right to your homescreen, in real time without opening the app.
You don't open Widgets!! In order to ADD a widget to your home screen, press and hold on the home screen until an ADD window comes up (press and hold is like a right click), then you choose WIDGET from the menu, and you should see a list of all the possible widgets you can add. Select the desired one and it will appear on your home screen.
If you have a lot of widgets running, they can use memory to stream information. Most widgets are configurable, but this is something to keep in mind. That being said I easily run 15 widgets on 5 home screens on my droid having them configured correctly, and can get more than a full day out of my battery.
Themes (and other Application add-ons)
Themes and other add-ons are programs that are used by the applications you have. Themes are the most popular so we will focus on those here, but there are others and you should get the idea. A theme typically runs on a home screen replacement, and changes the wallpaper, icons (for phone, browser, etc.), the drawer image, and other visual aspects of the interface.
99% of Add-Ons DON'T RUN!
Remember, these are add-ons for another applications, so they don't RUN like applications. 99% of the time, you have to OPEN them USING the program it's intended for. Open the program the add-on is intended for, and go to the MENU where you should find an option for THEMES. From there you can select from the themes you downloaded.
If your add-on isn't for themes, the process is the same. Open the application it's intended for, and find an option for the add-on in the application's menu. Select the add-on there.
IN MOST CASES, IF YOU READ THE DESCRIPTION OF THE PRODUCT IN THE MARKET, THE DEVELOPER WILL EXPLAIN HOW TO INSTALL IT, SO ALWAYS READ CAREFULLY.
I hope that helps some people with figuring out how to use all that the Market has to offer.