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:

1. The original OMTP configuration, which was created by Nokia and has a pinout of Left Audio, Right Audio, Mic, Ground.
2. AHJ (American Headset Jack) with a pinout of Left Audio, Right Audio, Ground, Mic.
3. 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.

Any headphone with a 3.5mm jack will work.

But be mindful about the remote controls. If it doesn't specifically says compatible with iPhone/iPod, it may not work with your iPhone.

As long as you just care about listening to sound on the headphones, as long as the tip is undamaged and standard, you won't harm the iPhone using any headphones.

Any headphones with a 3.5mm headphone jack will work fine with the iPhone, not just Apple products.

Sennheiser makes great headphones for iPhone with working mic/remote. I have the version for Android phones and everything works fine and sounds great. Just make sure you check the package for the "made for iPhone" label.

And note that some headsets have plugs with 4 metal bands, but the order is different, these won't work (older Nokia, for instance):

Instead of "mic, ground, right, left" they will have "ground, mic, right, left"