New to vector graphics?
You need vector graphics editing software to open a vector file and maintain the vector data. Common vector programs are Adobe Illustrator, CorelDraw, FreeHand and many more including free and open source. See Wikipedia list of graphic vector editors.
Open the appropriate vector program and from the menu select: FILE >> OPEN or IMPORT. If you simply double click on the file, your computer may find another program to open it and change the file's native format. Opening the file with a raster program like Photoshop will rasterize the image and the file will be of no use to anyone requiring vector art.
If you are not familiar with vector files and do not have a vector graphic editor but simply acquired vector art for a third party: a printer, sign-maker, etc., my suggestion is don't bother trying to open it, just send the vector file to your supplier and request a proof. Your supplier should give you a proof in a format that can be easily viewed.
I know what I am doing, I still can't open this %$#^&*! file
Then there is the compatibility issue. Not every vector program can open every vector file (e.g. an Adobe Illustrator ai file may not be opened with CorelDraw and a CorelDraw cdr file may not be opened with Illustrator) but most vector programs can import/open and export/save as eps file format. I believe this interchangeability has made the vector eps very popular throughout the years. And although I've read rumors that the eps may be replaced by the super-famous pdf, my itty bitty business has not seen a decline in demand - yet.
Can't work with the eps either? Since I am no tekkie I don't know the reason why but I do know that simply down-saving the file in an older version of eps (postscript level 3.0/adobe illustrator 8) has worked for my loyal patrons every single time, regardless of what vector editing software they are using.
How can you down-save an eps that you cannot even open? Ah, that's a problem. I would ask whomever provided you with the eps file to down-save to an older version, but often-times the origin of the eps file is unknown. 'Tis the reason I try to keep up with the latest software updates (which reminds me; I'm behind) - it can get very costly but avoids many headaches.
Good Night and Good Luck!