Can I use non-apple headphones with an iPhone? Is it okay to connect non-Apple headphones to an iPhone (a 5s, in this case) or do I need to use apple branded or authorized headphones for some reason?
Here is the visual comparison of Apple TRRS plug and a standard TRS plug.



On the top you can see a 4-pin headset connector. A headset connector needs to be using the same TRRS standard, or it will not function properly if at all.

On the bottom you can see a 3-pin headphone connector. Any headphone connector will work fine.



Standard 3.5mm jack stereo headphones have a 3-conductor configuration of TRS (Tip, Ring, Sleeve). The pinout for these connectiors is: Left Audio, Right Audio, Ground. Connectors for headphones with a mic have an additional ring which results in a 4-conductor TRRS configuration (Tip, Ring, Ring, Sleeve).

There have been different TRRS standards:


The original OMTP configuration, which was created by Nokia and has a pinout of Left Audio, Right Audio, Mic, Ground.
AHJ (American Headset Jack) with a pinout of Left Audio, Right Audio, Ground, Mic.
CTIA, named after The Wireless Association, which replaced AHJ and has the same pinout. CTIA jacks have improved compatibility to support not only AHJ but also OMTP. CTIA is by default required for Android device manufacturers.



Source: All About Windows Phone

Wikipedia has a short overview for which device uses which standard.

At last a few points to note:


Plugging an OMTP headset into an AHJ jack, and vice versa, will result in the audio-out being inaudible or very quiet. Converters to switch between the two types of headsets are readily available, but do add extra bulk, and may not work reliably with remote control functionality.
Apple uses a CTIA/AHJ connector arrangement, but uses a non-standard microphone and volume control signaling method. Simple commands like Play/Pause and Skip however should work across modern mobile devices.