I have lost an application I downloaded

Applications get lost a lot by people who are unfamiliar with Mac OS X's file system.  

If you properly install a software you download off the internet it should be in your hard drive -> Applications folder.

If it is not there, use Find Any File to locate it:

http://apps.tempel.org/FindAnyFile/

Note a very common mistake people used to Windows make is assuming the floating translucent bar on the bottom, known as the Dock, is a folder.   It is NOT.    The same thing with the left sidebar of the Finder windows.

They are merely a shortcut bars.    The path applications take from download to use should follow one of the following steps:

1. Download the .dmg file to your Users -> yourname -> Downloads folder, or a different folder you set in your browser  preferences.

2. Double click the .dmg file, so a virtual disk image appears on your desktop and left sidebar.

3. Inside that image will be either a folder with the application, or an installer. 

Here's where the possible steps diverge:

4. If the image contains a folder or application that is not an installer, drag it to Hard drive -> Applications folder.

4. If the image contains an installer, run the installer and make sure the program gets installed in your Hard drive -> Applications folder.

Once the installation is complete, you can drag the virtual disk image to the trash, and if you want to keep a backup of the installer, keep the .dmg file elsewhere.  

Sometimes the .dmg file, has .gz, or .zip attached to the file name. 

Older installers are .sit, .hqx, and require Stuffit Expander to open from http://www.stuffit.com/

If you inadvertantly dragged the application file to the Dock or double clicked it from within the image file itself, dragged the dmg file to the trash, quit the application, and emptied the trash, and never dragged/installed the application to the Application folder on the hard drive, you have lost the software, and will have to redownload it.  If it is a software you purchased from the Mac App Store, contact the App Store billing or the billing office where you downloaded the software from, to ensure you are not billed twice for it.

So you have to make sure you actually installed the software, and didn't just open the dmg file and application from the dmg file, before quitting, removing the dmg file, and emptying the trash of it.  

Otherwise, if the dmg file is not in the downloads folder, look for it in the trash and pull it back out to install the software afresh.