Android - Prevent apps being uninstalled

There are quite a few apps in the Market which will lock certain apps behind a password, a search in the Market for "applock" should find most of them.

I haven't actually used any, so can't give you a recommendation, but these should allow you to lock the Market and Settings apps behind a PIN number, to keep your kids out of them.

Install Smart App Protector with its helper app (for enhanced reliability). Make sure to make it device administrator. Then, lock Package Installer and Play Store using it (lock down other market apps too). With one tap, the app can lock all apps which could uninstall it. The app facilitates this for its own security, but its solution of your question.

And, I can bet... after that your kids can't uninstall an app without developer level expertise.

After your comment, looks like your app launcher is capable of uninstalling apps directly like Play Store. So, lock your Launcher too. From your screenshot, its Samsung TouchWhiz launcher having package name tw.launcher. Lock this package with Smart App Protector. But, it'd lock your app drawer too.
If you don't want to lock your app drawer, best way is to switch your Home Launcher app & lock Package Installer, Play Store etc. Done!
Alternate launcher apps have more features as well as memory efficiency. So, its better in all other aspects. If you don't want to spend money, I'd recommend you Go Launcher EX. If you can spend money for this, I'd recommend you ADW Launcher EX.