The Purpose of Buddha's coming to the world.

[icy]

The purpose of Buddha's coming to the world was for the sake of sentient beings' attaining the wisdom that he achieved. The paths that he taught are only a means leading to Buddhahood; he does not lead sentient beings with a low vehicle that is not a method leading to Buddhahood. He establishes sentient beings in the powers and so forth that exist in his own state.
"Manjushri, all the doctrines that I teach to sentient beings are for the sake of attaining omniscient wisdom. Flowing into enlightenment and descending into the Mahayana, they are means of achieving omniscience, leading completely to one place. Therefore, I do not create different vehicles."
-- from "Chapter of the True One Sutra".
--from Tantra in Tibet by H.H. the Dalai Lama

[Big Uncle]

There are 2 types of Buddhas, the wheel-turning Buddha called SamyaksamBuddha, in which they are compelled to teach because of the right mixture of the merit of the time and the compassion of the Buddha. However, there are some Buddhas due to lack of affinity and merit from our side, do not teach. They are called Pratyakabuddhas or solitary realizers. They gain enlightenment based on meditation and exit the world without teaching.

Here's a more extensive explanation that I found on net :-
Samyaksambuddha
Samyaksambuddhas (Pali: sammasambuddha) gain Nirvana by their own efforts, and discover the Dhamma without having a teacher to point it out. They then lead others to enlightenment by teaching the Dhamma in a time or world where it has been forgotten or has not been taught before, because a Samyaksambuddha does not depend upon a tradition that stretches back to a previous Samyaksambuddha, but instead discovers the path anew. In the Bahudh?tuka Sutta ("Many Kinds of Elements Discourse," MN 115), the Buddha tells Ven. ?nanda:
It is impossible, it cannot happen that two Accomplished Ones, Fully Enlightened Ones, could arise contemporaneously in one world-system—there is no such possibility.
The historical Buddha, Gautama Buddha, is considered a Samyaksambuddha. See also the list of 28 sammasambuddhas.
Three variations can be distinguished in the way of achieving Samyaksambuddha-hood. With more wisdom (prajñadhika), with more effort (viryadhika) or with more faith (sraddhadhika). Shakyamuni was a Prajñadhika (through more wisdom) Buddha. The next Buddha of this world, Maitreya (Pali: Metteyya) will be a Viryadhika (through more effort) Buddha.

Pratyekabuddha
Pratyekabuddhas (Pali: paccekabuddha) are similar to Samyaksambuddhas in that they attain Nirvana without having a teacher. Unlike the Samyaksambuddha however, they do not teach the Dhamma that they have discovered. Thus, they also do not form a Sangha of disciples to carry on the teaching, since they do not teach in the first place.
In some works they are referred to as "silent Buddhas". Several comparatively new Buddhist scriptures (of later origin; after the Buddha's demise, like the Jatakas), show Pratyekabuddhas giving teachings. A Paccekabuddha can sometimes teach and admonish people, but these admonitions are only in reference to good and proper conduct (Pali: abhisamacarikasikkha), not concerning Nirvana.
In some texts, they are described as 'one who understands the Dharma by his own efforts, but does not obtain omniscience nor mastery over the Fruits' (phalesu vasibhavam).

[Tenzin K]

Thus one of the benefits that Buddhism provides the human world is the knowledge of what is wholesome and what is unwholesome. This is a kind of right view which is a very important factor for your welfare as well as that of others, because only when you have right view will you know how to walk on the right path. For example, after listening to the Buddha, the stage manager Talaputa gave up his actor career and became a bhikkhu. Then he practised meditation and before long attained arahatship. On the contrary, he who has no right view will probably do whatever he likes, such as indulging in sensual pleasures, craving for name and fame, drinking, gambling. These will bring him suffering for a long time. On the other hand, a person with right view would likely engage in wholesome deeds, such as performing giving, practising virtuous conduct, cultivating loving-kindness and compassion, and purifying his mind through meditation. These will bring him happiness for a long time.

In the Dhammapada verses 316 and 317, the Buddha says:

“Those who are ashamed of what is not shameful, and unashamed of what is shameful, such beings, embracing wrong views, go to the woeful state.”

“Those who see fear in the non-fearsome, and do not see fear in the fearsome, such beings, embracing wrong views, go to the woeful state.”

These words of the Buddha are a real reflection of the modern age. For example, many poor people are ashamed of their poverty, and many rich people are so proud of their wealth; unattractive people are ashamed of their ugliness, and beautiful people are so proud of their beauty. But are money and beauty the yardsticks of what is shameful and what is not shameful. Certainly they are not. In either cases, if the person is virtuous, then there is nothing to be ashamed of, but if the person is immoral, then even if he is very rich and handsome, there is nothing to be proud of. Having known this you should always examine whether the thing you are going to do is wholesome or unwholesome. And through doing only what is wholesome, you become harmless to other beings; this is a kind of giving of fearlessness. Furthermore you yourselves also will gain benefits from your wholesome deeds.

[Midakpa]

Buddhas or enlightened beings appear in this world from time to time to convey the same message.  Buddhas do not have conflicting messages because there is only one Truth and they have realised it.  Therefore, out of compassion, they appear in this world to teach us to walk along the same path that they have walked in order to realise ultimate happiness or Buddhahood. By walking the path, it is possible to realise  our potential for awakening and becoming Buddhas just like them.

What is the Buddha's message? What did he teach? The 45 years of the Buddha's teaching can be summarised in the following words: "Do not do evil, do good and purify the mind." These are simple words which are easy for a child to understand but it takes many lifetimes to perfect.

[fruven]

A bit out of topic but this an interesting insight into pratyekabuddhayana told by Manjushri

From the Mahayana Sutra of Heroic Samadhi
Amidst the buddha's retinue were two hundred adherents of the bodhisattvayana who became discouraged, thinking, "Omniscience is hard to attain. We shall pass into parinirvana through the pratyekabuddhayana."

In order to subdue them, the youthful Manjushri said to the bhagavat, "In the past occurred the kalpa called Complete Radiance. During that kalpa I passed into parinirvana three hundred and sixty sextillion times through pratyekabuddhayana."

That retinue doubted this. Shariputra asked the bhagavat, "Since there is no continuation after nirvana, how could this be possible?"

The tathagata replied, "Ask Manjushri. He will tell you."

Shariputra asked Manjushri, "How are we to understand what you said?"

Manjushri replied, " The omniscience tathagata is my witness. Were I to declare otherwise I would be lying to the tathagata. During the kalpa of Complete Radiance a sambuddha called Victor appeared. He passed into parinirvana, but his holy dharma lasted for a hundred thousand years. After its disappearance, the beings there came under the power of conditions and could only be tamed by the demeanor of a pratyekabuddha. Even had a hundred or a thousand buddhas taught them they would have been uninterested; they were only interested in pratyekabuddhas. At the time there were no pratyekabuddhas to receive their offerings, so I declared myself to be a pratyekabuddha in order to ripen those beings. Wherever I went - villages, palaces, and towns - I was perceived as pratyekabuddha. Those beings respected, served, and supported me. As they gradually became vessels, I taught them dharma. I rose into the sky like the king of geese. They gained faith and made the aspiration to the pursue the path leading to such miracles. In such ways I fully ripened the virtues of many beings."

"When I felt that those beings had become tired of supporting me, I caused the words, 'It is time for me to enter nirvana' to be heard. Those beings came to me, carrying various offerings. I entered an absorption in which all perception and sensation ceased, and passed into parinirvana through the force of aspiration. They cremated my body and placed my ashes in a stupa built for that purpose. They venerated my remains and believed that I had passed into nirvana.

"In such a way I passed into parinirvana through the pratyekabuddhayana three hundred and sixty sextillion times during that intermediate kalpa, at the royal palace and elsewhere. By doing so I tamed thirty-six million beings by means of the pratyekabuddhayana. Although I passed into parinirvana through the pratyekabuddhayana, I never passed into complete nirvana."

......


After reading this text a few times it seems that the pratyekabuddha do teach when the condition as mentioned 'power of conditions' that sentient being only listen to a pratyekabuddha. One can also draw conclusion that there is only one buddha and many arhats. In the case of Complete Radiance kalpa there is one buddha but many pratyekabuddhas. What is more awesome is Manjushri became the pratyekabuddha! 

The second point worth mentioning is Manjushri said 'It is time for me to enter nirvana' and he enter parinirvana and returned! So one who enter parinirvana can choose to come back. Buddha Shakayamuni said the same thing as well. What happens after he entered 'nirvana'? Think about it.

[pgdharma]

A Buddha is a fully awakened being who has completely purified his mind of the three poisons of desire, aversion and ignorance and is no longer bound by Samsara and has ended the suffering which unenlightened beings experience in life. Out of  compassion, a Buddha reincarnate back to benefit sentient beings by sharing his insights to help sentient beings end suffering through eliminating ignorance by way of understanding and seeing dependent origination and eliminating cravings and thus attain the highest happiness, Nirva?a. His wish is to help everyone become a Buddha, to be free from Samsara.

[RedLantern]

T o follow the Buddha is to gain a clear and thorough understanding of this precarious human existence. Without this understanding,we will be like a ship sailing at random in a vast ocean from this shore towards a distant destination and such reckless sailing is extremely dangerous.Buddhism explains where life comes from,and where death leads to.It shows us what we are supposed to do now,in order to land safely on the other shore of light.
What is the basic purpose of following the Buddha?A Buddhist should understand that the significance and purpose of following the Buddha is to attain perfection.If we understand thoroughly the purpose in following the Buddha,we will feel confident that it is essential to follow the Buddha's teachings and tread a true path and learn the essence of Buddhism rather than being side-tracked or practicing incorrectly.
Buddha was a teacher and role model.He is anyone,who has become Enlightened,and anyone has the ability to become Enlightened.

[dondrup]

A Buddha would appear at a place and in a particular time due to the karmic affinity that the sentient beings have had with that Buddha. When sentient beings have exhausted the causes for the Buddha to remain, the Buddha will disappear into another realm of existence. 

All the one thousand Buddhas of this fortunate eon will appear in the human realm to perform the deeds of a Buddha.  Buddha Shakyamuni is the fourth Buddha of these thousand Buddhas that will appear.  The next Buddha to appear is Buddha Maitreya. 

All Buddhas of the past, present and future and of the ten directions have become Buddhas for the sole purpose of liberating every sentient being in samsara.  In another words, Buddhas will come to the World to turn the wheel of Dharma to show sentient beings the path to liberation and enlightenment.