VGA ports look like thishttps://en.wikipedia.org/wiki/File:SVGA_port.jpg
The ends of a VGA cable look like thishttps://images-na.ssl-images-amazon.com/images/I/71UDHnnbnaL._SL1500_.jpg
There's no way you could fit a VGA cable on a DVI port.
Now if your monitor requires an analog connection (DVD-I) and doesn't work with DVI-D (digital) connectors, you might get the connectors confused. I haven't seen a monitor that couldn't work with DVI-D in years though.
What model monitor do you have?
Here's more about DVI connectorshttp://www.datapro.net/techinfo/dvi_info.html
If your monitor can't accept a digital signal, you'll have to be very careful about what video card you buy. I think all the current model video cards are DVI-D like the one you just bought.
I am wondering why there was a big plug inside the computer that went to the video card and this one doesn't have that.
If you're talking about the extra connector, as seen on this card,https://images10.newegg.com/ProductImageCompressAll1280/14-487-244-15.jpg
the card you bought doesn't need that connector because it doesn't consume much power, so it gets all it needs through the PCI-E slot and doesn't need the extra connector. The card you're replacing consumed more power and it did need the connector.