Do you still need iPhone when running with Apple Watch?

You are able to use the Apple Watch when running without taking your iPhone with you. The caveat to do doing so is that you won't have access to any online services but the included fitness apps will work just fine!

This article at MacWorld has actually summarised exactly what you can and cannot do when your watch isn't connected to your phone. The paragraph that might be of most interest to you is:

If you want to go for a run without having to tote a bulky phone, the Apple Watch’s exercise capabilities are pretty much self-sufficient. All of the watch’s onboard sensors, such as the accelerometer and the heart rate monitor, keep on trucking even when your phone’s not around. Just fire up the Workout app, pick your exercise type and goal, and leave the iPhone at home.

References:
What can your Apple Watch do without your iPhone? | MacWorld