Can I charge my iPhone with my iPad charger?

TL;DR

Yes you can charge an iPhone with an iPad charger (and vice versa), and no it will not damage it.

THE LONG ANSWER

The term "charger" is often misused when referring to an iPhone/iPad "charger". Apple includes a power adapter with your iPhone/iPad, not a charger. The charger for an iPhone/iPad is inside the device itself. This internal charger is what's responsible for regulating the charging current to the device's battery - not the external power adapter.

With this knowledge, that the iPhone/iPad is responsible for regulating charging current, comes the understanding that using an iPad power adapter (or even a third-party adapter) will not damage your iPhone.

ADVANTAGES/DISADVANTAGES

iPhone 6/s or iPhone 6/s Plus

Charging your iPhone 6/s or iPhone 6/s Plus with an iPad power adapter (12W 2.4A) will not damage it. However, the iPhone 6/s and iPhone 6/s Plus were built with an internal charger that can take advantage of higher output. Therefore, charging your iPhone 6/s or iPhone 6/s Plus with an iPad power adapter will result in a faster charge than using the iPhone's power adapter (5W 1A).

Caveat: rapid charging Li-Ion batteries can result in the battery having a shorter lifespan. However, Stanford University recently completed a study that showed the effects of rapid charging Li-Ion batteries to be much less harmful than originally thought. Ergo, the negative impact (i.e., lifespan) of rapid charging an iPhone 6/s with a 2.1A (or 2.4A) charger vs. a 1A charger is negligible.

iPhone 5s or earlier iPhone

There is no advantage or disadvantage to charging your iPhone 5s, or earlier iPhone, with an iPad power adapter. These iPhone models were not designed to take advantage of power adapters with a higher output, so they won't charge faster. Like all iPhone/iPad devices, however, these models were designed to regulate the amount of current they need to draw, so using an iPad adapter will not damage them.

iPad

There is a disadvantage to charging your iPad with an iPhone power adapter: the iPad will take longer to charge when using a power adapter with a lower output. This will not damage your iPad, but it may frustrate you, which may damage your relationship.

Third-party power adapters

Using a third-party laptop, power strip, wall mounted, or car USB outlet to charge your iPhone/iPad will not harm your device, but it may take longer to charge if the amperage is lower than what the device can use.

Another word of caution when using multi-port third-party power adapters: the amperage is often divided between the number of ports the device has in use. So, if you are using a 2-port 1A (5W) USB outlet with only one device plugged in, that device will have the full 5W of output at its disposal. However, if you plug a second device into that USB outlet, the 5W is then divided between the two devices, resulting in each device having access to only 2.5W of output. Be sure to check with the manufacturer of the USB outlet to make sure you're getting what you think you're paying for.

FURTHER PROOF

Think all this is too good to be true? You certainly don't have to take my word for it. Take a gander at Apple's list of devices that are compatible with the iPad 12W Power Adapter. You'll note that all the iPhones appear on this list. It's pretty hard to argue with the manufacturer. Not impossible - but harder.

QUESTIONS/CURIOSITIES

Why does Apple ship the iPhone 6 and iPhone 6 Plus with a 5W power adapter if it can charge faster with a 12W power adapter? Your guess is as good as mine. But, I do have a few theories: 1) cost is always a factor when it comes to running a business, and 5W power adapters are less expensive to manufacture. 2) Li-Ion batteries do have a longer lifespan when charging them slower, insignificantly longer in my opinion based on Stanford's recent study, but maybe Apple engineers disagree. 3) Apple was sitting on a surplus of 5W adapters after manufacturing them for 7 years, and they wanted to get rid of them.

Perhaps, at the end of the day, Theory 1 + Theory 2 + Theory 3 = 5W power adapter in Apple's mind. We'll probably never know. But, we do know that we can charge any iPhone with any iPad power adapter without worrying about it, and we can rapid charge our iPhone 6 with an iPad power adapter and be all the happier for it!

Reference: http://2toad.com/Blog/2014/12/28/charge-your-iphone-with-an-ipad-charger