Hello!
Yoga serves one simple purpose and goal, that is easily described. It’s not about art, not about health, not about the divine, not anything like “Yoga just is”. Well. Yes it is. 
You can put it that way, if you want to or if you have to, but I (like “personally”) find it… cumbersome. The real explanation is much easier, easier to tell, easier to work with, and look, here it comes:
Yoga is a method to get the one who practices it to stop their thoughts. That’s pretty much it. You learn to stop thinking. Try it, try not to have anything going on up there, where you now mumble the words I’ve written. It won’t work, thoughts will arise immediately, instantly, within one, two, three seconds.
Yoga teaches you to stop those movements of the mind. It’s already pointed out in the first Yoga Sutras, catch a look, if you must, there are multiple translations online, you can find them with any search-engine.
All limbs of the Asthtanga Yoga (= Hatha Yoga) serve exactly that purpose, to get rid of thoughts and other mind-moving stuff:
Yama and Niyama, be good, so you’re reincarnated in a nicer place than now? Or end the terrible wheel of being reincarnated over and over again? Nice story, just that noone is around who really knows they’re reincarnated. The explanation is much simpler: If you do not do and think anything bad, there is no need to reflect your own behaviour, no need to feel guilty, nothing to worry, nothing to question, nothing to distract your meditation. How could you meditate, if yesterday you killed someone, or insulted someone, or thought bad about someone? Such behaviour will get back to you, even if you express your bad thoughts somewhat politely and without obvious insults, for which you could be sued. Don’t do and think anything bad, and no thoughts concerned with ethics and morals will bug you. That’s what these first two segments are for, not more, not less.
Asanas. Asanas aren’t intended for health, originally it’s even just the Asana, and it is a seated position that you can maintain for hours without having to adjust it, without feeling uncomfortable. The reason why you don’t just go to bed is, that you want to have your head upright, which is neccessary to keep a clear mind. That’s what the 3rd limb is intended for, and seriously: Nothing else. Asanas for health are a fine thing, sure, even good for the actual purpose of Yoga, as you won’t be able to stop thinking if your body is in pain. You might even encounter some trancendent effects, because you deeply concentrate while doing an Asana, but any other doing can create the same purpose, so there is nothing of great mystery in Asanas, no divine geometry that puts you in some line with the Great One. Use them for therapy or to get flexible, strong, a better athlete, a better car-seller, a better whatever. But within the main-frame of Yoga, Asana is to sit down, forget the body and practice stop-thinking.
Pranayama is to calm the mind via the breath, as breath and mind are related. Short breath, busy mind, deep breath, calm mind. And the other way. Pranayama intends to deepen the breaths, which is again intended to help one stop thinking.
Pratyahara: If you are pulled out of meditation by any signal coming through your senses, it sucks. So learn not to be bothered by any noise, smell, visual signal and no thoughts will arise due to these.
Dharana starts to train your mind to stop moving. As it’s very difficult to stop any process at all, you start with focusing the mind, preferably on a spot, or for beginners, on sounds (for example music, rivers, tides - water works very good, like anything monotone, steady, repetative), complex images (like Mandalas or a picture of some Guru), ideas (like one of God), points on or in your body (like the 3rd eye) or whatever, as long as it’s limited and the mind is kept in boundaries.
I do not dare to explain what Dhyana and Samadhi is, it’s both meditation, which is a deeper form of concentration. Here, you really stop thinking, stop concentrating the mind, but have the mind become one with an object or that “divine”, which has no form at all. This is the “fine art” of not thinking, which is very very difficult, but still some sort of exercise of the mind, just like playing the piano or dancing is an exercise of the body. Here you - as they say - can reach enlightenment, I’ve read a figure in several places and it says, that you will get there, if you stop thinking for about 30 minutes. One example is here, by Dharma Mittra:
"The old masters defined concentration as the ability to keep the mind on one point for twelve seconds without a break. Twelve concentrations - or two and half minutes -equal one meditation. Twelve meditations take a half hour. If you can concentrate without any break, if the flow of concentration is uninterrupted like oil pouring from a spout, then you’ve achieved the last state of yoga, samadhi, cosmic consciousness."
30 minutes. I also read this in a book by Swami Narayanananda, it should be “Secrets of Mind Control”, one very insight- and helpful book that I cannot recommend enough, particularly to those, who have problems to read through ancient scriptures, like the Gita and the Sutras.
That’s it. Yoga in a nutshell.
Many other - to go on a tangent - spiritual methods serve the same purpose. For example to believe in a god and have faith and trust, pray to god, keep your mind always occupied with god. This reduces your thoughts, because there is nothing to think about, nothing to worry about, if you hand over your whole life to god, live your life by gods rules, don’t question what happens to you (read Hiob). Praying is Dharana, particularly that… where you have that little thingy with the littly thingies on them (lol), you know, where you pray the same prayer over and over again. It might be called a rosary or so. There are quite a few Dharananic practices, like not speaking for a long time: If you won’t talk to anyone anyway, so many thoughts become obsolete. Even beating yourself up for bad thoughts and desires is some sort of self-education. The whole ascetic thing, no sex, no luxury, that all serves that purpose of reducing thoughts, for if you manage to get rid of those desires (instead of raping children, feeding yourself fat and building churches out of gold), you have so much less to think about.
Is there really a god? Someone watching over everything, punishing these, gratificating those? I’m not buying it, just like I’m not buying reincarnation. These are vehicles, methods. Good methods, though, and if one choose it, of course they have to stick to it, so rational debate makes no sense at all. Swami Narayanananda explains a lot about this method in the mentioned book, according to him you can even create a god so vivid, that you can see, hear and touch them (him or her). Some form of self-hypnosis I guess. This method also is used by some living Gurus who provide themselves to be believed in by you, there’s a woman quite famous, forgot her name, but I think someone on this board is using her for that purpose. [Edit: This is her, ‘Shri Mataji Nirmala Devi’.] Unfortunately, not everyone is made to have that faith-ability. Unfortunately, because the world of the west only has that method and it leaves a lot of people frustrated with religion.
Speaking of religion: Find out what the word actually means, it comes from latin “religare”, look it up and compare it’s meaning with the meaning of the word “Yoga”, that has been explained in this thread already.
Yoga, though, has more methods, there are four types of people: Those who have faith, those who like doing stuff, those who can concentrate very good and those who think a lot, from the Wikipedia:
Karma Yoga: The yoga of action in the world [do-people]
Jnana yoga: The yoga of Wisdom and intellectual endeavor [think-people]
Bhakti Yoga: The yoga of devotion to God [faith-people]
Dhyana Yoga: The yoga of meditation [concentration-people]
Any individual is mixed of all four types, but usually one is obviously dominant (for me, for example, it’s the thinking, I am a highly intellectual person, and faith doesn’t work very well for me; with a “regular” religion only, I’d be screwed). Yoga serves all types of people and provides methods for everybody, that’s why there are so many forms of Yoga and not just Bhakti, which is the Yogic Christianity.
Well: Nuff said!
You can put it that way, if you want to or if you have to, but I (like “personally”) find it… cumbersome. The real explanation is much easier, easier to tell, easier to work with, and look, here it comes: