Assuming you have Xcode installed in /Applications, then you can do this from the command line to start the iPhone Simulator:

$ open /Applications/Xcode.app/Contents/Developer/Platforms/iPhoneSimulator.platform/Developer/Applications/iPhone\ Simulator.app

(Xcode 6+):

$ open /Applications/Xcode.app/Contents/Developer/Applications/iOS Simulator.app

You could create a symbolic-link from your Desktop to make this easier:

$ ln -s /Applications/Xcode.app/Contents/Developer/Platforms/iPhoneSimulator.platform/Developer/Applications/iPhone\ Simulator.app ~/Desktop

(Xcode 6+):

$ ln -s /Applications/Xcode.app/Contents/Developer/Applications/iOS Simulator.app ~/Desktop

As pointed out by @JackHahoney, you could also add an alias to your ~/.bash_profile:

$ alias simulator='open /Applications/Xcode.app/Contents/Developer/Platforms/iPhoneSimulator.platform/De‌​veloper/Applications/iPhone\ Simulator.app'

(Xcode 6+):

$ alias simulator='open /Applications/Xcode.app/Contents/Developer/Applications/iOS\ Simulator.app'

(Xcode 7+):

$ alias simulator='open /Applications/Xcode.app/Contents/Developer/Applications/Simulator.app'

Which would mean you could start the iPhone Simulator from the command line with one easy-to-remember word:

$ simulator

There is a recently announced service Corellium that offers virtualised cloud instances of iOS. It's not publicly available yet though and no word of how much it will cost. I'm not aware of any way of running them on your desktop though other than the Xcode simulator.

It is the kind of thing I imagine Apple would have been quick to clamp down on in the past although there is lots of talk about the two app stores converging in the near future so who knows what will happen.

See these links for more info on that:

• https://9to5mac.com/2017/12/21/apple-cross-platform-apps-opinion/
• https://www.bloomberg.com/news/articles/2017-12-20/apple-is-said-to-have-plan-to-combine-iphone-ipad-and-mac-apps

UPDATE:

Apparently a number of researchers have had varying degrees of success booting iOS in QEMU. Here is an article providing a guide on getting iOS 12 to the point of launchd and recoveryd running:

https://worthdoingbadly.com/xnuqemu2/

In it's current state it will not allow you to run apps but very interesting none the less and worth keeping an eye as Corellium still appears to be invite only.

UPDATE 2:

Apple is attempting to slowly destroy Corellium. The user daeken on Twitter has been developing an emulator for iOS simulator on macOS that they have stated they are close to releasing (May 2020) though, but will be a commercial product. They appear to have put in a huge amount of work and have demonstrated a number of apps working on it.