Which DSLR I can buy for Photography?

Be aware that installing Magic Lantern voids your warranty.

Any of the three cameras you list will do a great job of shooting kids, family, indoors, outdoors, friends, but hide if some bride wants you to take wedding pix.

You really need to take yourself to a camera shop and handle both a Nikon and a Canon. While you're there look at the Sony and Pentax cameras also.

Just to make a decision, I'd go with the Nikon D5100 and the 18-105mm kit lens.

All these cameras are great. The canon t3i is known around the world for its great video mode, however the nikons have a slight advantage in photo mode. The D3200 has 24MP and the D5100 16MP, both are good enough to achieve great photos. But megapixels does not necessarily equal better images, most people actually prefer d5100 quality over the d3200, even though its older. All of them have mic inputs, so thats good for video. All in all, it all comes down to which you prefer. I recommend you visit a local camera store, and pick up all cameras and take pictures and scroll through the menus etc. You will find some are easier to use and feel better. I would recommend Nikon D5100 because the interface is more user friendly, for example it shows you the aperture and the light meter etc. With canon you need to know what your doing. Nikon also has a higher frame rate, allowing to capture fast moving subjects (kids,sports etc), and much shorter buffer times. You might want to invest in lens in the near future, nikon usually have cheaper glass also. Canon t3i is for advanced video.

OVERALL- The nikon d5100 also has a twist screen like the t3i. D3200 lacks that. I would suggest you to stay away from the d3200.