Make Your Theme Your Own
After you have installed your brand new theme, things might be looking a little less like you expected them. Fret not! This just means there are a few more steps to take.
One of the first things you'll need to do after installing a new theme is to regenerate your image thumbnails. Each theme has pre-set image dimensions for images so that they fit correctly within a theme's area. Whenever you install a new theme, you should run the Force Regenerate Thumbnails plugin to resize your images. For detailed instructions on how to install a plugin, see this tutorial.
Navigate to Appearance > Widgets. First things first, some widgets might have moved around - from example going from the Sidebar to the Header Right area, etc. Go through every Widget Area listed and ensure it contains the right widgets. If not, drag and drop the widgets into their proper areas.
Additionally, each theme demo makes use of widgets in specific ways. Be sure to read the theme documentation for your specific theme to get answers on how to set up your blog to look like the demo.
Set Featured Images
It's possible even after regenerating thumbnails and setting up your widgets that you will see spaces that don't look quite right. If you haven't been doing so, you might need to set a feature image for your Pages and/or Posts. You can do this by going to the specific Page or Post, clicking Edit, and adding a featured image in the Featured Image box on the right hand side of the post editor.
After all these things are complete, visit your specific theme documentation. This will answer specific questions pertaining to your theme such as setting up menus, uploading your logo, and customizing colors.