Ahh, so much conversation.
Anyways, I have a few things to add. Nikon and Canon have don't have built-in vibration reduction/image stabilization in their camera bodies and autofocus in cheap bodies. There's also certain lenses that are compatible with each camera. Nikon DSLR lens compatibility chart:
http://www.kenrockwell.com/nikon/compatibility-lens.htm#dslr
Canon discontinued their FD mount and replaced it with the EF mount in the 1980s for autofocus. Obviously, the mounts aren't compatible. I think, but I am not completely sure, that Canon's full frame lenses aren't compatible with their APS-C bodies, vice versa. This is a nuisance compared to the Nikon F mount which has been around since the Nikon F.
There's two sensor sizes that are used in DSLRs nowadays (APS-C and 35mm), both based on film. The cameras with 35mm sensors are much more expensive due to the more advanced technology to utilize a larger sensor and to cut out bigger silicon wafers.
The larger the sensor, the better the camera's image quality and noise performance. The Sony RX100 is the leader in compact cameras due to its 1" sensor, much larger than all pocketable point-and-shoot camera sensors. I still use my Canon Powershot S90 more than my DSLR due to its portability, but there is a definite downside in terms of image quality with its 1/1.7" sensor (~1/3 the surface area of Sony's similarly sized RX100). I've easily noticed a lack of quality in my 16 x 24" and even 8 x 12" prints compared to prints from my DSLR.
I chose the S90 in particular over four years ago mostly due to its ability to shoot RAW (having manual control was necessary and a wide aperture was desired). All DSLRs have this capability (ignore 12-bit vs 14-bit and compressed vs uncompressed), but not even all high-end point-and-shoots can output these files. If you're serious about photography, go with a compact that can shoot RAW (Panasonic LX7, Olympus XZ, Sony RX100, Fuji X10/X20) or a DSLR. Nikon makes bad point-and-shoots and Canon's S90-S110 line became mediocre. If you don't care for portability, go with a DSLR. They have much larger sensors and much better performance. But DSLR lenses and accessories can make this choice easily the most expensive.
The only benefit mirrorless cameras seem to have is their small size. If you're interested in a cheap interchangeable lens compact (ILC), I'd consider Sony then Olympus and Panasonic.
Recommendations:
Compact: Sony RX100 ($650, but unmatched image quality)
DSLR (entry-level): Nikon D3000 series, Canon 500D series, Pentax K30 (tons of pro features for cheap)
Mirrorless ILC: Sony NEX series, Panasonic G series, Olympus PEN series
(There's also cameras with large sensors but fixed lenses like the Sony RX1 and Fuji X100S.)
EDIT:
As Cras mentioned, the most important thing about a camera with changeable lenses are the lenses. If you're going to save up for a good body (D7100) and have no money for lenses, it's almost pointless. In two years, your camera body will be outdated yet the lenses generally keep their value and can serve you for decades. Sure, the D7100 offers more features and performance than the
much cheaper D3100, but unless you know that you need these benefits, it's a bad idea to spend more on the D7100 body.
Also, check this website out:
http://snapsort.com/compare