Can I use normal headphones in the iPhone 5S jack?
The headphone jack on the iPhone 5s can be used by any set of headphones with a standard 3.5mm jack. It will not damage or affect either the headphones or phone in any way.

The only difference is that some headphones have buttons that use resistance between the signal and ground wire in the headphone cable control the phone. That's what your Apple Earbuds are doing. Not having this on the headphones is perfectly OK.

Of course if you have a look around you can find excellent headphones that do have the buttons. I have two different pairs of headphones with good sound, a microphone and phone controls.