Having just downloaded Aperture
Is that the first time you are using Aperture, or are you using an older Aperture library? If it is an older library, how has it been created?
If it is a new installation - is there a crash log with the error message? If yes, could you please paste the first 50 lines of the report here?
Usually the crash log will be shown directly in the panel with the error message - look for a text starting like this:
Process: Aperture 
Version: 3.4.5 (3.4.5)
Build Info: Aperture-320090000000000~1
Code Type: X86-64 (Native)
The Console.app (in Applications > Utilities) will show you all log files stored on your computer.
Press "Show log list" in the Toolbar, and then look for diagnostic reports in the side bar.
Sounds like you have a possible library or permissions problem.
You wrote that this is a new installation, have you imported anything into the Aperture library yet?
Two things you can try, one start Aperture while holding down the command and option keys. This will bring up the library first aid window. Run the permission repair after it is done if the library still doesn't open start Aperture the same way again (command and option keys) and run the library repair.
You could also try making a new library. Start Aperture while holding down the option key only. This will bring up the library selector window. Select the option to make a new library and see if Aperture will run with that.