I understand that there are some Christian groups that believe that it is possible for a Christian to live a life entirely free from sin.
What is the nature of this belief and what is the basis for it?
I believe your question is in reference to my group Pentecostal Holiness and Word of Faith. We believe that we believe in our heart unto righteousness. I will attempt to explain this doctrine. (but please do not down vote me to infinity if you want to call yourself a sinner). Here I've selected scriptures and comment on the scriptures and Strong's definition. I reference the scriptures by key words.
The rightful desire of every believer is to walk in the righteousness and Holiness which is ours in Christ. We do it by faith. Faith means believing, speaking, acting and thanking God for the things which He has provided for us in His word. Faith means acting counter to what is factual sometimes. The fact may be that you belong in Hell but the truth of the matter is the while you were dead in sins Christ died for you making you worthy. It is up to you to walk worthy.
We would no sooner call ourselves a sinner than we would cuss the reason being is that we are do not believe that we had a dual nature as many other Christians believe. We believe that our spirit has been made one with the Lord and that we are new creations in Christ. We do believe that contented with the flesh and to a limited extent with devils however we put the emphasis on what we can do rather than what can affect us. Since we are new creatures we believe that we have been created in Christ. To be in Christ means to partake of His righteousness, holiness and justification. Just as Christ is holy in Him so are we. Christ is without sin and in Him we have been justified (or made just as if we had never sinned).
Sinners need to experience the blood of Jesus. Those cleansed by the Blood of Jesus are cleansed perfectly and consciousness are purged from law of morality and Mosses but then turn in service and do the things which please the living God. We repent a whole lot and after every repentance we consider the scripture to be true when it says we have been purified. Needing repentance doesn't mean we are practicing sin and being sinners it simply means we failed to do what we could do and we must do better longer with God's help next time.
To be filled with righteousness to walk in the fullness of what Christ provided for us through his redemption is to be mature in Christ. Being mature in Christ to us is an obtainable non static Goal.
We do not count ourselves to have righteousness of ourselves but clothe ourselves in Christ. That we may obtain what has already obtained us. It has been said that God's Grace holds the believer and keeps them. We equally believe that we can hold onto God and grasp his goodness in our lives being fully mature in all that He has to offer.
In commanding us to be perfect (or fully mature would be a better translation) Christ did not set an unobtainable goal. To say Christ told us to do something which was not possible for us to do is to call Christ a liar.
We believe the key to walking in full maturity and the nature of Christ and God is to walk in the God kind of Love rather than yielding to human emotions. This is a kind of brain washing or renewing of the mind begins when we are little. Other believer are taught as children to memorize the 10 commandments and do Bible memory verses about specific knowledge from or of God (orthodoxy) instead we teach some orthodoxy but major on orthopraxy. The reason for this is that though the word be perfect we have only a fraction of what there is to know. We access that which is known unknown by following the Holy Spirit being loving Rom 2:14. Children likewise begin when they are little being corrected by their parents and Bible teachers that their actions do not or do conform to the standard of God's Love given in 1Co 13. It is not uncommon for a child never to memorize the Lord's prayer because instead they are taught to just have a conversation with their loving heavenly Father (later in life in their own studies they learn the lesser Bible points).
Our culture is different than other Christians. You have to understand since we have been striving to walk in love and all it entails since we were children. We tend to view those who have problems with walking in love as children and refer to them as spiritual children. We don't mean disrespect and we will teach you how to act it just will be harder for you since you are an adult.
Not everyone chooses to follow this path as no one ever said it would be simple it is however easy with time. Those that do have had many years of training on how to respond in themselves and to others in a way which isn't sin not that they always do but the idea is that they always can. In commanding us to love God and love others Christ gave us a practicable and doable commandment. Those who follow those two commandments will automatically do anything righteous besides them.
Also Galatians 5 plays an important role in our worship. We believe that being born again and living in fellowship with the Holy Spirit of God is different than walking in the Spirit of God. That if we grow and display the fruit of the human/holy S/spirit (which are designed to help other people) we will be able to not give into our flesh. In resisting temptation we suffer. Likewise we suffer not doing other optional things like fasting or special offerings. In suffering we are able to train ourselves to cease from sin. This is not suffering being sick, broke or in any other anguish of the body or mind but a choice to follow what we perceive as the will of God as revealed by the scripture.
To us the so called sins of omission are not real sin at all. Only that which is a violation of "light" or knowledge given from God is a sin intentionally walking in darkness is sin but ignorance in knowledge or action is not a sin. If you choose to stay ignorant after receiving the Gospel (or even basic truths contained in the gospel) that is a sin and you are liable for the continuance of that sin and the sins which you do as a result of that sin all your life.
The motivation behind our strives towards perfection is to look into the face of the Master and hear "well done good and faithful servant". We do not believe that all believers will hear this on the day in which all believers are tried by fire. Our goal is to have as little part of our lives be burned and suffer as little amount of loss when the fire try's it so that we can receive the maximum reward to throw at the throne of Jesus a few minutes later.
For further reading along this line: The Two Kinds of Righteousness by E. W. Kenyon, Understanding Righteousness by Kenneth E. Hagin, Reigning in Life by Kenneth E. Hagin, Understanding Righteousness by Mark Hankins, Separate from Sin by Keith Moore, Keeping the Love Command by Keith Moore, God is Love by Keith Moore, Love's Sacrifice by Keith Moore, Growing Up by Keith Moore, Growing Up Spiritually by Kenneth E. Hagin.
I think you are referring to a doctrine called Entire Sanctification, which is taught by some Wesleyan and holiness churches. A Google search on that term will give you more information, but basically, the idea is that it is possible with the Holy Spirit's help to avoid sin completely. Some Scriptural basis for this belief can be found in Romans 6, 1 Thessalonians 5:23, and 1 John 2:1-2, which talk about Christians being free from sin and blameless.