If Adam sinned and fell, why should I need salvation? How can the sin of Adam affect me? What does the Psalmist refer to when he says my mother conceived me in sin? Please explain the concept of natural sin clearly.

These are difficult and important questions and deserve clear, biblical answers. I will do the best I can to help you understand these matters and trust the Holy Spirit to be our teacher as well.  

The failure of Adam to resist the temptation of the serpent in the Garden of Eden had a catastrophic effect not only on coming generations of mankind, but also on all of creation. Because of the Fall, "the whole creation groaneth and travaileth in pain together until now" (Romans 8:22For we know that the whole creation groaneth and travaileth in pain together until now.
See All...) and is in the "bondage of corruption" (v.21). We can easily observe the world around us and human society in particular and see that something is terribly wrong. The Fall of Adam as recorded in Genesis Three explains what went wrong.

However, I understand that in our age of personal rights and individualism it seems unfair that we should be penalized for the failure of another. I will seek to demonstrate that we actually accept this as a common state of life, but I will also seek to show the biblical logic behind this representation by Adam. I will also emphasize that no one goes to hell simply because he or she is an ancestor of Adam. In order to explain this doctrine, I am going to show three ways in which we are affected by sin and what the particular results of each way is.

SINNERS BY REPRESENTATION

Theologians call this the federal headship of Adam, but I do not like this terminology. In this doctrine, Adam stood in as our representative. I think it may be easiest to understand this by thinking of Adam as our champion, much as Goliath was the champion of the Philistines when David fought him and killed him. Goliath challenged Israel to send their champion so that the two of them might fight. He declared, "If he be able to fight with me, and to kill me, then will we be your servants: but if I prevail against him, and kill him, then shall ye be our servants, and serve us" (1 Samuel 17:9If he be able to fight with me, and to kill me, then will we be your servants: but if I prevail against him, and kill him, then shall ye be our servants, and serve us.
See All...). In victory or in defeat, Goliath was standing in for the entire Philistine army.

We have no trouble understanding this in sports and war. If my team wins, then we have won. If the army of my nation wins, then I share in the victory. On the other side, if they lose then so do I. In this, God was not tricking us or setting up a scheme to defeat us. Rather, He created a perfect man, Adam, and placed him into a perfect environment, Eden. Daily, God walked with Adam in the cool of the day in order to instruct him and draw close to him. He made a world in which man could do as he pleased without fear of disobedience. Adam was created in a state of innocence, meaning that he had no disposition toward evil but gladly did the right and obedient thing. I believe that Adam was superior to any man who has lived since his creation. God put the best man in the part of our champion.

Then, in order to give man a chance to prove his love to the Lord and His willingness to submit to Him, God gave Adam one and only one prohibition. He was not to eat of the tree of the knowledge of good and evil but was to trust in the Lord to reveal His knowledge in His time (Genesis 2:16-17 [16] And the LORD God commanded the man, saying, Of every tree of the garden thou mayest freely eat: [17] But of the tree of the knowledge of good and evil, thou shalt not eat of it: for in the day that thou eatest thereof thou shalt surely die.
See All...). Since the command to be fruitful, multiply, replenish the earth, and subdue it was given before sin entered the world (Genesis 1:28And God blessed them, and God said unto them, Be fruitful, and multiply, and replenish the earth, and subdue it: and have dominion over the fish of the sea, and over the fowl of the air, and over every living thing that moveth upon the earth.
See All...), we can conclude that if Adam had refused to take of the forbidden fruit, that he would have multiplied and lived in an earthly paradise. He would have won paradise for all who would be his descendants. If this had been the case, I doubt that we would have complained that we were the benefactors of his obedience. However, his disobedience lost paradise for us.

Before we go further, let me introduce the key Bible passage on the disobedience of Adam:

Romans 5:12Wherefore, as by one man sin entered into the world, and death by sin; and so death passed upon all men, for that all have sinned:
See All... Wherefore, as by one man sin entered into the world, and death by sin; and so death passed upon all men, for that all have sinned:  13 (For until the law sin was in the world: but sin is not imputed when there is no law.  14 Nevertheless death reigned from Adam to Moses, even over them that had not sinned after the similitude of Adam's transgression, who is the figure of him that was to come.  15 But not as the offence, so also is the free gift. For if through the offence of one many be dead, much more the grace of God, and the gift by grace, which is by one man, Jesus Christ, hath abounded unto many.  16 And not as it was by one that sinned, so is the gift: for the judgment was by one to condemnation, but the free gift is of many offences unto justification.  17 For if by one man's offence death reigned by one; much more they which receive abundance of grace and of the gift of righteousness shall reign in life by one, Jesus Christ.)  18 Therefore as by the offence of one judgment came upon all men to condemnation; even so by the righteousness of one the free gift came upon all men unto justification of life.  19 For as by one man's disobedience many were made sinners, so by the obedience of one shall many be made righteous.

Adam, as our representative head, brought sin and corruption into the world. This sin brought death--the curse of all men. 1 Corinthians 15:22For as in Adam all die, even so in Christ shall all be made alive.
See All... states, "For as in Adam all die, even so in Christ shall all be made alive." This leads to the second way in which sin affects us. We are:

SINNERS BY NATURE:

When Adam sinned, his very nature was corrupted. He lost his innocence. The man who walked with God hid from Him in the garden. Instead of accepting his guilt, he passed the blame on to his wife. As such, he received the judgment of God on his life and flesh (Genesis 3:17-19 [17] And unto Adam he said, Because thou hast hearkened unto the voice of thy wife, and hast eaten of the tree, of which I commanded thee, saying, Thou shalt not eat of it: cursed is the ground for thy sake; in sorrow shalt thou eat of it all the days of thy life; [18] Thorns also and thistles shall it bring forth to thee; and thou shalt eat the herb of the field; [19] In the sweat of thy face shalt thou eat bread, till thou return unto the ground; for out of it wast thou taken: for dust thou art, and unto dust shalt thou return.
See All...). Though Adam was created in the image of God (Genesis 1:27So God created man in his own image, in the image of God created he him; male and female created he them.
See All...), his son Seth was born in Adam's likeness and image (Genesis 5:3And Adam lived an hundred and thirty years, and begat a son in his own likeness, and after his image; and called his name Seth:
See All...). The perfect image of God was now corrupted and that corrupted image and likeness was passed on to all the descendants of Adam.

One of the most significant elements of this corrupted nature is the natural leaning of man toward disobedience; his natural state of rebellion toward God. The scriptures are filled with testimonies to this fact. The heart of man is deceitful above all things (Jeremiah 17:9The heart is deceitful above all things, and desperately wicked: who can know it?
See All...). All are filthy and there is none that does good (Psalm 14:1-3 [1] The fool hath said in his heart, There is no God. They are corrupt, they have done abominable works, there is none that doeth good. [2] The LORD looked down from heaven upon the children of men, to see if there were any that did understand, and seek God. [3] They are all gone aside, they are all together become filthy: there is none that doeth good, no, not one.
See All...). Even the righteous acts of man are as filthy rags (Isaiah 64:6But we are all as an unclean thing, and all our righteousnesses are as filthy rags; and we all do fade as a leaf; and our iniquities, like the wind, have taken us away.
See All...). We all inherit this natural tendency to sin. David's confession--"I was shapen in iniquity; and in sin did my mother conceive me" (Psalm 51:5Behold, I was shapen in iniquity; and in sin did my mother conceive me.
See All...)--refers to this sinful nature. David recognized that from his conception it was his nature to sin.

Again, we were represented by Adam and in his loss of innocence we lost. This may be compared to the biblical teaching that Levi was in the loins of Abraham and when Abraham tithed to Melchisedek, Levi also tithed to him (Hebrews 7:9-10 [9] And as I may so say, Levi also, who receiveth tithes, payed tithes in Abraham. [10] For he was yet in the loins of his father, when Melchisedec met him.
See All...). In the same way, we were in the loins of Adam when he rebelled against God and lost his pure nature. Therefore, we now participate in the results of that action.

However, I want to make it clear what we lost in Adam and what still falls to our own responsibility. In Adam, we lost a home in a perfect earth (paradise on earth); we lost eternal life (we now die); we lost a perfect innocent human nature (we have a corrupted fallen nature that tends to sin). However, Adam is not responsible for our individual sins. We sin by choice. Neither are we condemned to hell because we are descendants of Adam. It is our own sins and our own rejection of God's way of salvation that condemns us to hell. Though we lost much in Adam, we are still responsible for our own woes. This leads to the third way in which sin affects us. We are:

SINNERS BY CHOICE; SINNERS BY PRACTICE:

The passage cited earlier tells us that "death passed upon all men, for that all have sinned" (Romans 5:12Wherefore, as by one man sin entered into the world, and death by sin; and so death passed upon all men, for that all have sinned:
See All...). Our death and condemnation to hell is the result of our own sins. We are still responsible for what we do with God's commands. God gave the law as a way to life. The problem was that no one obeyed it. Paul stated, "Is the law then against the promises of God? God forbid: for if there had been a law given which could have given life, verily righteousness should have been by the law" (Galatians 3:21Is the law then against the promises of God? God forbid: for if there had been a law given which could have given life, verily righteousness should have been by the law.
See All...). The law could have given life if men would have obeyed it. But they did not find life this way because of the weakness of the flesh (Romans 8:3For what the law could not do, in that it was weak through the flesh, God sending his own Son in the likeness of sinful flesh, and for sin, condemned sin in the flesh:
See All...).

However, if the law could have given man eternal life, then men were not utterly condemned because of their ancestor Adam. He put them in a bad position, but the law could have brought them out. Yet, they still failed to save themselves. In this manner, every man is equally guilty with Adam and equally responsible for their own sin and destruction. As sinners by representation, we lost the earthly paradise. As sinners by nature, we lost the original innocent nature. However, it is only as sinners by practice and choice, that we are condemned to an eternal judgment in hell.

Second question is: If the sin of Adam affected my soul without my permission, why does the salvation of Christ not cleanse me without my permission? Why do I need to accept Christ to receive salvation? Why does the salvation Christ offers require my involvement when the sin from Adam affected me without any of my involvement? Is the sin of Adam more powerful and suppressing than the salvation of Christ? I am genuinely confused with these questions. I will be much happy and grateful if you clarify my doubts.

This second question is basically answered in the first. I am not simply a victim to the disobedience of Adam. I am responsible for my own actions as well. Even with the curse brought on by Adam's disobedience, I was not condemned until I committed my own willful sins of disobedience and rebellion. Therefore, since I am not simply a sinner without permission, I cannot be a child of God without permission. I must submit my own rebellious heart to the Lord Jesus and trust in Him and the provision He has made for my forgiveness and for eternal life.

This long post has still only touched the surface of this important Bible topic.