How to know which camera lense will fit my camera?

I have had a Nikon D40X and Recently both my lenses were broken (grrr children!) And so I need to invest in new ones.

How do I know a lense will fit my camera?

Also which type of lense would you recommend? I mainly use it to take photos of family parties and carnivals aswell as the occasional landscape/wildlife photo

Your D40 takes motorised lenses in Nikon AF fit - any Nikon lens will fit, any third-party lens in Nikon mount will fit, but if you want autofocus, check for a motor (Nikon call them AF-S, Sigma say HSM, and Tamron and Tokina will have similar abbreviations).

I'm assuming that your lenses would be the AF-S 18-55mm, and probably the AF-S 55-200mm, as they would be the ones included in a twin-lens kit from Nikon or a participating dealer. In any case, it will do you no harm at all to simply replace the lenses you've lost.

All Nikon cameras from the 60s and on have what is called an F mount. So all lenses made for Nikon meet that standard.

If you like the lenses you had, just order ones just like them. If you want to try something different, get Nikkor (Nikon) AF-S lenses or motorized third party lenses from Tamron, Tokina, or Sigma. You would probably be disappointed with cheapos from Rokinon, Vivitar, and the like. Just be sure you order third party lenses with the Nikon mount.

Since a lot of people upgrade, there are probably used 18-55mm AF-S VR lenses available in very good condition.
Some day you may want to upgrade your camera body. The new bodies have 24 MP. A lens that looks good at 6 MP might not look so great at 24 MP. The 18-200mm AF-S VR II could be an example of that.
The Sigma 17-50mm f2.8 HSM OS would probably continue to give good service on a new24 MP body.

You let your children use/play with your delicate photo equipment?!
Why not just buy the same lenses you had which were broken?

A decent salesman at a decent store or the internet will tell you.

Any Nikon F mount lens will fit your camera (ie any lens made by Nikon since about 1959).

Nikon's AF lenses won't autofocus though, so you'd need to manually focus the lens. If you want autofocus, look for Nikon AF-S lenses or Sigma HSM lenses (Nikon fit of course!).