The account of Jesus's encounter with the rich young ruler is given in Matthew 19:16-26, Mark 10:17-27, and Luke 18:18-27.
This encounter is frequently used to teach that we have to keep the commandments to go to Heaven, and so it's important to have a good understanding of what Christ is teaching here.
Notice that in the question the young man asks Christ, he starts with the presupposition that he must do something good in order to get Eternal Life:
Matthew 19:16
16 And, behold, one came and said unto him, Good Master, what good thing shall I do, that I may have eternal life?
Mark 10:17
17 And when he was gone forth into the way, there came one running, and kneeled to him, and asked him, Good Master, what shall I do that I may inherit eternal life?
Mark 18:18
18 And a certain ruler asked him, saying, Good Master, what shall I do to inherit eternal life?
As can be seen by what the man says, particularly in the Matthew account, he has set in his mind the idea that he must do a "good thing" to get to Heaven.
Christ's response immediately begins to chip away at the young man's misconception. He tells the young man that only God is good:
Matthew 19:17
17 And he said unto him, Why callest thou me good? there is none good but one, that is, God: but if thou wilt enter into life, keep the commandments.
Mark 10:18
18 And Jesus said unto him, Why callest thou me good? there is none good but one, that is, God.
Luke 18:19
19 And Jesus said unto him, Why callest thou me good? none is good, save one, that is, God.
What was Christ's intention in saying the above? If "there is none good but one, that is, God", by implication, the young man is not "good". Yet, what was the young man looking for? A "good thing" that he should do, to have Eternal Life.
Then, Christ tells him to keep the commandments:
Matthew 19:17-19
17 And he said unto him, Why callest thou me good? there is none good but one, that is, God: but if thou wilt enter into life, keep the commandments.
18 He saith unto him, Which? Jesus said, Thou shalt do no murder, Thou shalt not commit adultery, Thou shalt not steal, Thou shalt not bear false witness,
19 Honour thy father and thy mother: and, Thou shalt love thy neighbour as thyself.
Mark 10:19
19 Thou knowest the commandments, Do not commit adultery, Do not kill, Do not steal, Do not bear false witness, Defraud not, Honour thy father and mother.
Luke 18:20
20 Thou knowest the commandments, Do not commit adultery, Do not kill, Do not steal, Do not bear false witness, Honour thy father and thy mother.
Above, Jesus Christ is essentially telling the young man to "be good", after having told him that there is "none good". He has also given the young man the genuine method that someone can use to get to Heaven by doing a "good thing" - which answered the original question the man had asked.
If someone keeps the Law perfectly, they will earn their own way to Heaven, as Romans 2:13 also testifies. However, as we go on to read, since "all have sinned, and come short of the glory of God" (Romans 3:23), all have broken the Law (1 John 3:4). Therefore, we have to be "justified by faith without the deeds of the law" (Romans 3:28), and "no man is justified by the law in the sight of God" (Galatians 3:11).
The young man doesn't understand what Christ has showed him so far, and responds with the lie of "all these things have I kept from my youth up":
Matthew 19:20
20 The young man saith unto him, All these things have I kept from my youth up: what lack I yet?
Mark 10:20
20 And he answered and said unto him, Master, all these have I observed from my youth.
Luke 18:21
21 And he said, All these have I kept from my youth up.
Of course, the young man probably kept them to a degree - but so does everyone. The essential thing is that that degree is less than perfection, and so he's still not understanding the absolute standard that Christ is preaching to him.
Christ then responds to his question "what lack I yet?" with a commandment that He knew targeted a blatant sin in the man's life:
Matthew 19:21
21 Jesus said unto him, If thou wilt be perfect, go and sell that thou hast, and give to the poor, and thou shalt have treasure in heaven: and come and follow me.
Mark 10:21
21 Then Jesus beholding him loved him, and said unto him, One thing thou lackest: go thy way, sell whatsoever thou hast, and give to the poor, and thou shalt have treasure in heaven: and come, take up the cross, and follow me.
Luke 18:22
22 Now when Jesus heard these things, he said unto him, Yet lackest thou one thing: sell all that thou hast, and distribute unto the poor, and thou shalt have treasure in heaven: and come, follow me.
Here, Christ again uses the Law against him - covetousness is tantamount to idolatry (Colossians 3:5, Ephesians 5:5), and therefore breaks the First Commandment (Exodus 20:3).
This succeeds in convicting the young man, and "he went away sorrowful: for he had great possessions" (Matthew 19:22, Mark 10:22, Luke 18:23). The implication is that he was sorrowful because he now realized he wasn't good enough to keep the Law, but there is no indication that he understood that he needed Jesus Christ to save him.
After the young man departs, Christ turns to His disciples, and says:
Matthew 19:23-24
23 Then said Jesus unto his disciples, Verily I say unto you, That a rich man shall hardly enter into the kingdom of heaven.
24 And again I say unto you, It is easier for a camel to go through the eye of a needle, than for a rich man to enter into the kingdom of God.
Mark 10:23-25
23 And Jesus looked round about, and saith unto his disciples, How hardly shall they that have riches enter into the kingdom of God!
24 And the disciples were astonished at his words. But Jesus answereth again, and saith unto them, Children, how hard is it for them that trust in riches to enter into the kingdom of God!
25 It is easier for a camel to go through the eye of a needle, than for a rich man to enter into the kingdom of God.
Luke 18:24-25
24 And when Jesus saw that he was very sorrowful, he said, How hardly shall they that have riches enter into the kingdom of God!
25 For it is easier for a camel to go through a needle's eye, than for a rich man to enter into the kingdom of God.
He remarks on how hard it is for a rich man to be saved - and from Mark's account, we see that the issue is a misplaced trust - "how hard is it for them that trust in riches to enter into the kingdom of God" (Mark 10:34).
This is because every lost person - the rich young man being a prime example - has their trust in something other than Jesus Christ to justify them. The belief that takes place at Salvation is a transfer of this trust from oneself to Christ (Ephesians 1:12-13).
The disciples, themselves not understanding what just took place, are prompted to ask, "who then can be saved"? This leads us to the linchpin of the entire encounter:
Matthew 19:25-26
25 When his disciples heard it, they were exceedingly amazed, saying, Who then can be saved?
26 But Jesus beheld them, and said unto them, With men this is impossible; but with God all things are possible.
Mark 10:26-27
26 And they were astonished out of measure, saying among themselves, Who then can be saved?
27 And Jesus looking upon them saith, With men it is impossible, but not with God: for with God all things are possible.
Luke 18:26-27
26 And they that heard it said, Who then can be saved?
27 And he said, The things which are impossible with men are possible with God.
What is "impossible" for men? Salvation. On your own merits, you cannot get to Heaven. It's "impossible".
This is why we have to be "saved through faith" (Ephesians 2:8), "not by works of righteousness which we have done" (Titus 3:5). That is the entire point of the message, and the crux of the encounter - you are not good, you never will be good, you don't keep the commandments, and it's "impossible" to be justified without Jesus Christ.
Those who would use this passage to teach that a person gets to Heaven by keeping the commandments, are themselves the rich young ruler. They've failed to understand what Christ is saying in the exact same way that he did.
The point of this passage is to show someone their need for the Savior by confronting them with the demands of the Law. Christ is using the Law, as He does elsewhere, to make sin "exceeding sinful" (Romans 7:11-13), which was the entire purpose of the Law (Galatians 3:19, Romans 3:19-20), with the intended effect of ultimately leading one to Christ (Galatians 3:24).