This answer is for people who are just doing a general search for how to center things using a storyboard and don't understand constraints.
I assume you are using a Storyboard (not XIB files) and Auto Layout. This is the default nowadays so if you don't know what that means then don't worry about it. If you want to check, though, you can click Main.storyboard in the Project Navigator and then the File inspector. You can make sure that Use Auto Layout is checked:
Select your button (or whatever view you want to center) on the storyboard. Then click the align button on the bottom right. Select "Horizontally in Container" and "Vertically in Container". Click "Add 2 Constraints".
If it wasn't perfectly centered already you may need to do one more thing. Click the "Update Frames" button that is two to the left of the align button. Your view should now be centered on the storyboard.
And more importantly, now when you run your app it should be centered no matter what device size you are using.
I originally learned this from Creating the user interface Auto layout. The basic idea is still the same, but the UI arrangement has changed a little in Xcode now.
I realize that this does not answer your question. You were right to accept @Fogmeister's answer, not mine. However, because of your title many people come here expecting to learn how to center something on a storyboard. It is those people I am trying to help.