I John 2:1 and 2:
2:1My little-children, I write these-things to you in order that you may not sin. And if-ever someone may sin, we have an encourager towards the Father: Jesus Christ (the) righteous-one, .2and he is (the) propitiation concerning our sins, but not concerning our (sins) only but also concerning (the sins) of the whole world.
John continues from chapter 1 by addressing the holy-people (sanctified-people, Christians, saints, children of God) to whom he is writing as "my little-children" in the figurative usage being a very tender, affectionate term of endearment.
My little-children, I am writing these-things to you for the purpose and result that you would not sin (err, miss the mark of God’s Word to you, swerve from the truth of what God says to/for you).
And (also, even) if it should happen at any time that someone would commit a sinful action…
The Greek word translated "propitiation" comes from the Septuagint Greek translation in Exodus 25:17 as being the mercy-seat or cover of the ark on which the blood was sprinkled as the means of propitiation, which is the gaining or re-gaining of favor, blessing, atonement, etc (also refer to Leviticus 16:15). God purposed Christ Jesus to be both the sacrificial offering shedding his blood and the covering-place of propitiation, and thus God could provide the availability of propitiation to mankind (refer to Romans 3:25 and Hebrews 2:17).
Notice also that these verses refer to God as our Father. Even if we holy-people would commit a sinful action, we still retain the relationship of a Father (God) with His children (us); however, we would not be sharing-in-common with our Father at that time (refer to I John 1:3, 6 and 7).
When John writes "Jesus Christ" he is emphasizing his obedience and service to God his Father on a horizontal human level to benefit people. Jesus who is the resurrected Christ continues today to serve his Father.
The word "Jesus" is explained in Matthew 1:21 by the Angel Gabriel when he spoke to Joseph:
Matthew 1:21:
.21She [Mary] will bring-forth a son and you [Joseph] will call his name Jesus, for he will save his people from their sins."
"Jesus" means "God saves / Jehovah (Yahweh) the Savior," and the Lord Jesus Christ was given that name "Jesus" because he is the one who carried-out God's will to make it available for people to be saved, to be made-safe. It is by means of Jesus, the Christ, that people are able to be saved and thereby have a true and vital spiritual relationship with the only true God. Jesus Christ is God's son – he is alive today! He is now at God's right-hand side, second-in-command to God his Father.
The word "Christ" means the "anointed-one" or "Messiah." In the lands and during the time-period of the writing of the Bible, holy oil was poured on the high priest of God to begin his ministry for God, and it was also used to anoint kings. Many people and things were anointed with holy oil to signify their making holy or sanctified for a specific purpose, etc, which can be read about in many scriptures in the Bible (refer to Exodus 30:22-33; Acts 2:36, 4:27, and 10:38).
[Reference: John 1:29, 3:16, 14:16 and 26, 15:26, 16:7; Acts 3:14 and 15, 7:52, 22:14; Romans 3:25, 8:26, 27 and 34; Philippians 2:1; I Timothy 1:15; Hebrews 2:9, 17 and 18, 9:5; James 5:6; I Peter 3:18; I John 4:10 and 14.]
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True Bible Study - First, Second, and Third John
