When you are attempting to open a file created by Adobe Illustrator with another program, there are two primary considerations: the file type and what you're trying to do with the file. Illustrator can save projects using a variety of file types, each of which have different editing and viewing capabilities. How you choose to open the file will also depend on whether you are looking to edit specific elements of the file or simply view the image.
Opening an AI File
AI -- Adobe Illustrator -- is the default file type for Illustrator which saves completely editable and scalable vector graphic images. Certain AI files are viewable with PDF readers; however, they will not allow you to edit the file's layers or individual elements. When an AI file is saved, there is an option that users can select which creates a PDF representation within the file. This option also allows you to open an AI file in other Adobe applications, such as Photoshop.
If the file has been saved with PDF compatibility, then you should be able to view the file using any PDF viewer. To see if this is true, try opening the file with your default PDF application, such as Adobe Reader.
Editing an AI File
If you're looking to edit the file, a free alternative to Illustrator is Inkscape with many vector graphic editing features, such as hiding or rearranging layers or color fill and stroke tools. CorelDraw is another heavy hitter in the design world which offers many of the same features as Illustrator (some of which you won't find with Inkscape, like image resizing), but it also comes with the same hefty price tag. Check out this comparison of the Inkscape and Corel Draw features.
Another file type commonly used in Illustrator is EPS -- or encapsulated PostScript -- which contains both vector and bitmap graphics. EPS is a more generic file type not associated with a particular application, which contains graphic images that can be opened by a variety of graphic editors. They will not transfer the same editing data as AI files, since they are not a native file type.
Other File Types Related to Adobe Illustrator
As we've just explored, Illustrator is a versatile application that can save files in a number of formats. See all file formats associated with Illustrator. Illustrator can also be used create bitmap graphics, such as PNG or JPEG images, which can be used online and in print. Bitmap graphics will not contain any editing capabilities, but can be viewed in any web browser or image repository software, such as Windows Live Photo Gallery.