What is the equivalent Nikon model of the Canon EOS 600D and 650D?

Also which is better? Nikon or Canon. Can't choose between the two. I'm just beginning photography and I want the best DSLR.

Nikon D 3200 Nikon D90 etc are excellent choices.

Nikon D5200 is the parallel Nikon to the 600D and newer 650D

Here is how their sensors compare

http://www.dxomark.com/...nd3)/Canon" class='ext_link'>http://www.dxomark.com/...nd3)/Canon

A Nikon shooter will tell you Nikon is better - they will tell you about a review that proves it. For the equivalent Nikons, look at the D5100 and D5200.
A Canon shooter will point to reviews of output, rather than reviews of sensors, that proves otherwise, or highlight that Nikon don't make the sensor in the D5200, they get Toshiba to do that.

They end up arguing, and you end up without the help you need.

If you are unsure about what camera to use, ask people you know, whose output you have seen and you like.
Take a look on flickr.com at photos taken with the cameras, you can ask the photographers what they think of their cameras.
Then find your local camera store and go there. Talk to the people working there. Try different cameras. Find the one that suits you best.
You might find that it's not a Canon or a Nikon, but a Sony (who make sensors for Nikon as well), or some other brand.

Neither is 'better' or 'best' - both Canon & Nikon (& Sony, Pentax & Olympus) make camera models with similar features & quality at similar price points.

Decide on your budget, then pick the models from Canon/Nikon/any of the above that you are interested in, then go to a camera store & try them out. Check button position, menu choices, general 'feel' & weight & pick whichever one is best for YOU.

ANd yes, the Nikon D5200 would be equivalent to those Canon models.

The equivalent Nikon to the EOS 650D in price point is the Nikon D5200. You can see a comparison between the three cameras here:

http://www.dxomark.com/index.php/Cameras/Compare-Camera-Sensors/Compare-cameras-side-by-side/(appareil1)/850%7C0/(brand)/Nikon/(appareil2)/813%7C0/(brand2)/Canon/(appareil3)/692%7C0/(brand3)/Canon

The snapsort comparison between the D5200 and EOS 650D is here:
http://snapsort.com/...ikon-D5200

The Nikon is the newest of the three and has the most impressive stats, Nikon took a lot into account in bringing it out, an addition to the lineup only released late last year