Once you have picked a new theme for your Magento store and downloaded its installation package, you need to log in the Magento admin panel. Then go to System -> Magento Connect -> Magento Connect Manager.
This will open the Magento Connect Manager, from where you can install your new theme.
In the Direct package file upload section, click the Browse button and navigate to the theme installation package you have downloaded from the theme vendor.
Then you need to click the Upload button to upload the installation package to the server.
The installation will begin and your new theme will be installed automatically. Once the theme is installed return to the Magento Admin Area and open System -> Configuration -> Design -> Themes. Enter the theme name in the Default field and click on the Save Config button.
That's it! Now your Magento store should start using the newly installed theme.