A filename consists of two parts: basename, extension. For example the file notepad.exe has a basename notepad and a extension .exe. The file extension is used to identify what kind of file it is. With no extension there can be no direct association. So, you have to know exactly what the file's format is.
Why my downloaded file does not have any extension?
In most case, it is the fault of the webmaster. Users have nothing to do with it. You may download the file from the webpage again .
I know what the file is!
If you are quite sure what kind of file it should be, that is you know the proper file extension, things become simple. Just rename the file, and add the appropriate file extension and double-click to open it. To rename a file, right-click it, type the new file name, and then press ENTER.
NOTE: if you know it is a image but not sure if it is BMP, GIF or JPG, just add .JPG extension and try to open it. It should work.
I DO NOT know what the file is!
What do you do with a file that has no extension and you do not what it should be? Punt is good and safe choice.
Or if you may try to open it with different application like notepad, 7-zip to try your luck. When you double click a file containing no extension, the Open With dialog opens. In the Open With dialog box, you can click the program where you want the file to open, or click Browse to locate the program that you want.