Jesus' remarks were that those who would receive his teachings would be like little children. Also see the Beatitudes, which provides more detail about the simple childlike faith (my words) of those that would receive him (meek, pure in heart, peacemakers, etc.).
The Kingdom of God belongs to those who can accept Christ in the same way a little child simply believes an elder.
On the topic of original sin, Protestant teachings (although there are obviously exceptions) from approximately Martin Luther onward are along the lines that Christ's blood atones for children until the age at which they are capable of making a conscious decision to accept or reject His Grace. This is commonly referred to as the Age of Accountability. The doctrine of Age of Accountability is that there comes a point in your life when you must make a choice for which you will be judged (see Romans 14:12); the actual point in a child's life at which one becomes capable of decision making is not specified in scripture, and in my opinion would differ for each individual. Prior to this age, Christ's blood atones for "original sin" having paid the price "once for all" (Hebrews 10:10).
Romans 14:12 -
So then every one of us shall give account of himself to God.
Roman Catholic teachings, on the other hand, states that those who die without first being baptized remain under original sin, and therefore descend to hell (or Limbo, in some interpretations).
Council of Lyons II: “…The souls of those who die in mortal sin or with original sin only, however, immediately descend to hell, to be punished with different punishments…-- (Denzinger 464)
Council of Florence: “…Moreover, the souls of those who depart in actual mortal sin or in original sin only, descend immediately into hell but to undergo punishments of different kinds.— (Denzinger 693)
Pope John XXII: “It (The Roman Church) teaches… that the souls… of those who die in mortal sin, or with only original sin descend immediately into hell; however, to be punished with different penalties and in different places.” (Denzinger 493(a).
"It has been decided likewise that if anyone says that for this reason the Lord said: "In my father's house there are many mansions"(JN14:2): that it might be understood that in the kingdom of heaven there will be some middle place or some place anywhere where the blessed infants live who departed from this life without baptism, without which they cannot enter into the kingdom of heaven, which is eternal life, let him be anathema. For when the lord says :"Unless a man be born of water and the Holy Ghost, he shall not enter into the kingdom of God"(Jn3:5), what Catholic will doubt that he will be partner of the devil who has not deserved to be a co-heir of Christ? For he who lacks the right part will without doubt run into the left" (Denzinger 102 fn.2; 30th edition)
Note that current Roman Catholic church doctrine makes allowance for the possibility of cleansing from original sin for unbaptized children. Source.
In the documents of the magisterium in the Middle Ages, the mention of “different punishments” for those who die in actual mortal sin or with original sin only (“As for the souls of those who die in mortal sin or with original sin only, they go down immediately to hell, to be punished, however, with different punishments") must be interpreted according to the common teaching of the time. Historically, these affirmations have certainly been applied to unbaptised infants, with the conclusion that these infants suffer punishment for original sin. It must be observed however that, in a general way, the focus of these Church pronouncements was not on the lack of salvation for unbaptised infants, but on the immediacy of the particular judgment after death and the assignment of souls to heaven or hell. These magisterial statements do not oblige us to think that these infants necessarily die with original sin, so that there would be no way of salvation for them.