I've recently come by a dell Inspiron and upon booting it up found that the graphics card was fried and everything worked but was blue, the monitor was perfectly fine as with the cable, in suggestions for a new card?
I guess you can call the VRAM on a graphics card "storage." I typically think of storage as non-volatile space like a hard drive, a flash drive, or a disc. VRAM is volatile, thus when you turn off the PC, all the information stored is lost.
Anyways, to answer your question, the size of VRAM does and doesn't matter. If you are looking for high resolutions (greater than 1080 ie 1920x1080), more VRAM is needed. For gaming on resolutions below 1080, it's quite normal and acceptable to have VRAM in the 1 to 2GB range. The game itself is also important aspect. For example, Need For Speed and Battlefield 4 are going to have different requirements. For NFS, a 1GB card would work fine. But for BF4, I'd highly suggest getting a 2GB as it can and will use most of that VRAM.
The most important aspect is balance. If the Dell PC you have has a below average processor and little RAM, there's no point in buying a $200+ card because in the end, the PC itself may not be able to run the game in the first place.
Does the Dell have onboard graphics? If so, take out the GPU and connect the monitor to the onboard graphics and see what happens.