What is the difference between External martial arts and Internal martial arts?
This is not the best answer, though it is an answer.
"Internal" and "External" can be traced through Sun Lu Tang's writings on the "Neijia". Being of the Chinese literati, he used the words 內 (internal) vs. 外 (external) because it has double meanings. Besides the connotations of "esoteric" and "exoteric", those words were widely used to describe "domestic" and "foreign." Some martial arts express Taoist (that is, "domestic") philosophical principles. Some martial arts express Buddhist principles. Since Shaolin, like Buddhism, came from India, this makes Shaolin and Shaolin-derived arts as "foreign" arts.
When you dig deeper into the histories, it gets murkier. But in general, the Chinese are proud of being Chinese and like to point out things that originate from the Chinese as evidence of a superior civilization.
When American practitioners came back from China and Taiwan, the connotations of "domestic" and "foreign" got lost in the translation. The connotation of "esoteric" and "exoteric" got exaggerated out of proportion. And now the Internet has amplified this confusion.
I'd spend less time worrying about exact definitions of "internal" vs. "external" and more time in practicing the art. "External" arts will eventually take you through the "internal", and "internal" arts will eventually express through the "external." Don't cheat yourself.
Update 2014/02/19: To clarify the controversy, I'm well aware that the sense of "internal" vs. "external" in use, particularly in America, refers to some notion that external arts tend to use brute physical strength, and internal arts use whole body motion or perhaps refined with qi (氣). There is further, an argument that English is a living language, and should reflect the actual usage. That is a fair point. However, in this case, the popular usage of "external" vs. "internal" is a misconception -- a distortion that happened when it was taken out of context of the original culture and further distorted for marketing purposes.
For example, Shaolin (少林) is considered many by Western practitioners as an "external" art in the sense that it seems to use a lot of brute physical motions. This is a misconception. There is quite a bit of whole body motion training, ground path, and qi development. In other words, if you used "internal" vs. "external" to designate power generation in a lineage, this distinction is fuzzy. It is fuzzy because that distinction only appears in the minds of people who are marketing their art, either as teachers trying to attract students, or students trying to posture.
As another example, from what I've seen with Systema practitioners, their body movements and power generation expresses many of the principles found in what Western practitioners would call "internal". My encounter with Systema is fairly shallow, but I have never heard a Systema practitioner talk a big deal out of "internal" vs. "external" or posture about it. These folks train hard.
Further, if you investigate the origins even further, you'll find that what we now considered "internal" as far as the woo goes, that is, the neigong (內功) was traditionally taught as separate practices. It was not unknown for say, a Xingyi (形意) or a Shaolin practitioner to learn neigong from a different lineage. Neigong as a practice has never been restricted to a martial art, and so there are no martial art for which there is an inherent practice of neigong. It was only when the literati like Sun Lu-tang practiced it, they also did what they did best as one of the educated elites: documented, categorized and talked about it, and perhaps befuddled it.
So the neigong, what could arguably distinguish "internal" from "external" was never a distinguishing factor of a particular lineage. It was a distinguishing factor of the individual practitioner.
In other words: there is no such thing as an "internal art" or an "external art". However, there are individual practitioners who might express an art "internally" or "externally". If we want to use the Western sense of "internal" vs. "external", then use it distinguish the skill attainment of a practitioner rather than the art itself.
If you want to use it to distinguish lineages, the original distinction is a lot more useful ... if you are Chinese or Taiwanese and want to be proud of your homegrown Taoist lineage, defend the honor of your civilization, and all that crap.
Update 2014-04-10: I've seen some new things. I don't have anything coherent to share yet. I would have preferred to delete this answer outright but StackExchange will not allow me to delete accepted answer. Instead, I've made edits to say that this is an answer, not a bad story to tell about on this subject, and not the final word on the subject. I'm tempted to just erase the text, but people can check revision edits so there is no point.
External martial arts focus on concussive impacts. These impacts cause damage by hitting with more force. Training focuses on developing strength, speed and specific techniques.
Internal martial arts focus on using the leverage of the relaxed body to control an opponent. Training focuses on internalising (making second nature) a holistic set of attributes including correct posture, control over the tension-relaxation of muscles and correct positioning of the limbs and body without focusing on specific techniques.
I believe that internal-external is a spectrum, rather than absolutes.
An external martial art is one in which the emphasis is on physical application of force (whether your own or the opponents force redirected) to cause damage to your opponent. This is the case whether you are relaxed (re-directed force or joint locks etc.) or applying force directly via the application of brute strength.
An internal martial art is one in which the emphasis is on application of chi energy in order to disable the opponent. Once again, this is regardless of whether you are relaxed or applying what looks to be a strong, physical movement.
Internal martial arts stress the training of chi via exercises like Chi Gong and External martial arts stress the training of physical/muscular strength. This generalisation is true particularly at beginner levels. Later on both schools of thought tend to come back closer to the centre more often than not.
The above info is the classical definition of an Internal or External martial art.
There are also hard and soft styles. For example an external martial style can use redirection and deflection instead of applying direct opposing force, without it being thought of as internal. Similarly an internal style can apply what appears to be direct application of force to combat an attack, once again without being thought of as external.
The distinction between Internal and External martial arts I believe is primary around kung fu and other Chinese martial arts and was a way to categorize the distinct styles of kung fu that existed.
An internal kung fu is one with progressive and slow forms that allow for almost a moving meditation. They are much more of a finesse art. Your focusing more on your soul, mind and chi to control and opponent and not relying brute strength or force. Using principals of rotation, deflection and evasion to overcome an opponent, making them beat themselves. A good example of an internal kung fu would be Tai Chi.
An external kung fu is more of an outward showing of force and aggression, can be characterized by sharp, fast and succinct motions. Moving with deliberate motions and with purpose. In an external style you will have direct attacks, your will beat your opponent with your force and will oppose your will on theirs. An example of an external style would be Wing Chun.
I think the distinction is pretty blurry, and possibly moot, now a days. A lot of martial arts have elements that one could categorize as either internal or external.
External and internal martial arts are the same thing, but they start from a different place. All martial arts follow different paths up the same mountain, but the end result is the same. Because a teacher can't take you all the way up the mountain does not mean the style is incomplete.
The limited point of view is this:
In truth, here is what is most likely accurate:
It is the same mountain. Just a different path.
External MA is about explosive power. Think of a sprinter. Boom! He goes OUT from the gate. Internal is about implosive power. Think of a figure skater. Watch as she brings herself IN towards her core in order to spin faster.
may help clarify
Try reading Doug Wile's 3 tai chi books "Tai chi touchstones", "Lost tai chi classics from the late ch'ing dynasty" and "Tai Chi Ancestors", all available from Amazon. Wile traces the first use of "internal" to (From the Lost Classics book)
You may also find this blog post by Mike Sigman of interest where he covers a lot of ground in relation to internal arts.
The Neijia mailing list, which was the vehicle created by Mike Sigman in the late 1990s to promote serious development of the internal martial arts, defined the internal martial arts to be those that respected the six harmonies (Sigman 2012a, 2012b).
The list followed the classical definition of the six harmonies due to (Dai Longbang 1750), a master of Xingyi:
(Dai Longbang 1750) The Six Harmonies Fists, verse translation from http://www.centrosanti.com.ar/articulos/ver/the-6-harmonies
(Sigman 2012a) Silk Reeling, aka Six Harmonies Movement
External: the force is generated by local muscle.
Internal: the force is generated by the whole body, in which your waist and hips are the key part you need to coordinate well.
Any martial arts that can effectively use your whole body to generate the power of punching, grasping, etc., can be considered as an internal martial arts, or at least shares the same base of internal martial arts.
For high level practitioners, internal martial arts means you can not only generate power with your whole body, but also can change the direction, the amplitude, the position quickly and easily, up, down, in, out, left, right, front, back, clockwise, counter-clockwise, etc. Once you know how to do that, how to apply that to different scenarios, you know Yin-Yang, or in Tai-chi, you know how "jin" works