Hmm, it's complicated.
Basically, the DSLR is built around a CMOS sensor (the electronic equivalent of the film frame. The higher end DSLRs have a full size CMOS (i.e. same size as a frame of 35mm film). The rest will vary. A smaller CMOS means that your lens is effectively magnified (sort of like digital zoom).
With Nikon:
FX series - full frame
DX series - 1.5 ratio, so an 18-105 zoom lens becomes the equivalent of a 27-157.5
An FX lens can be mounted on a DX camera, but not the other way around (they will fit, but at the wide angle setting you will get "vignetting", which is the lens seeing the edges of itself, resulting in black corners on your photos)
The Canon equivalent is the EF lenses and the EF-S lenses (the EF-S are designed specifically for the smaller CMOS).
Unless you have very specific needs, like shooting under harsh conditions, there is no real need to go up to a full frame CMOS camera, which would be considered high-end.
Full frame pros:
Image quality (but realistically, most non-pro now are over 20 megapixel, which is more than enough)
Higher ISO range (for shooting in low-light without flash)
Weatherproofing (my 1d can be taken out in the rain for instance)
Shutter and FPS speed (if you need 1/8000 of a second or 10 frames per second shooting) The latest 1D does 14 frames per second, crazy stuff.
Durability, longer lifespan, if your are planning to shoot many thousands of exposures.
Endurance (more shots per battery, as the batteries are generally bigger, which is also a con)
Full frame cons:
Weight. My 1D weighs 1.25 kilo. Without lens. I'm upgrading soon to a 5D (downgrade?) due to weight. My wife wants to play photography and the 1D is just too heavy. The new 5D has way better features that my older 1D though.
Cost. Unless you really need that performance, you're just paying too much.
Useability. The pro cameras dont have the cool features like setting for "sport" or "portrait". It's lot harder to just point and shoot.
So, the D5200 or 3200 both have great specs. Its over 24 megapixel, does facial recognition, HD video.
The most common lens is the 18-55, or the 18-105 if you want more reach, at the price of more weight ans size. And more price.
Cheers.