I don't keep up with tech so much these days so I have a question.
 
My computer (a Dell Inspiron 3668) has one PCIe x16 slot and two PCIe x1 slots. I have a Delta 44 PCI soundcard I need to use in my computer. I've talked to Newegg twice and was told "no worry, the PCI card will fit the PCIe x16 slot". Let me know if this is correct and if not is there a adapter I could use (my soundcard if full-height)? Thanks in advance.

PCI slots (older) and PCIe slots (x16, x8, x4, and x1) are completely different.
Do not try to put your Delta 44 into a PCIe x16 slot.
It will NOT work.
 
If your motherboard doesn't have a PCI slot, the Delta 44 can't be used.

there are some PCIe to PCI adapters that may work. newegg and Amazon have some. some are very expensive ($200+) and others seem to simply be some buffers ($30). I haven't used them but I have a delta 66 and delta 1010lt both of which need PCI (32-bit) and most newer MOBO only have the PCIe types...

Some of the lower priced boards still have them.  M-Audio PCI cards have no issues for me so far.

The OP's motherboard (from the explanation)... does not have a PCI slot.
 
Current motherboards that do have PCI slots:
Those PCI slots are bridge into the PCIe bus via a bridge-chip.
Some audio interfaces don't work with bridged PCI slots (Lynx, Echo)
Other audio interfaces generally work fine with bridge PCI slots (RME, M-Audio).