Verses 1-2. Here Paul lays out an "easy" way to show respect to all in the assembly. He tells Timothy not to sharply rebuke an older man, but to appeal to him as a father. He then applies these paternal terms to other members of the assembly (younger men & women, Older women). This instruction is easy to remember, though difficult to practice. It is difficult to practice because while the instruction to not sharply rebuke carries over to all 4 age/gender combinations, so does the directive "in all purity." Note that Paul is not saying not to rebuke others (see v. 20), he only restricts the manner in which it is done and the spirit (pure) in which it is done.
The benefits of a gentle rebuke are obvious -- psychology would tell us that the approach is key when trying to correct poor behavior. However, the real challenge is to remain pure. Many times have I tried to correct a brother or sister. Most of those times I have failed to approach the offender in an appropriate manner. The rest of the time (or so it seems), I had impure (read: selfish) motives. I trust that God's will was accomplished, however imperfect His tool was.
Verses 3-16. Immediately, Timothy has to put this respectful rebuke into action. The issue of the widows is a case study which Paul uses to illustrate his point. Timothy is instructed by Paul to honor the widows who are widows, and correct an issue with widowhood which was prevalent in this assembly. He had to tell them what they were doing was wrong and then give them alternate behaviors which would be more acceptable. (Note that a sinful act is always replaced by a holy one.) All the while, he had to treat them as mothers/sisters and in all purity.
People hate to be corrected. Timothy was not in an enviable position at all. Even if Timothy did everything right, he would probably still endure harsh, unholy criticism from the congregation. However, leaders in the assembly must be subjected to that from time to time for the sake of the glory of God.
Verses 17-25. But while the work is hard, Paul says the reward is great as we remain faithful to God. Verse 21 is a key when considering the goal of leadership: maintain God's principles and don't be partial. In other words, respect God while respecting fellow man. Shall we take it so far as to say follow the greatest two commandments, Love God & Love Man? That would seem right.
If we keep those two strait, then the rest will follow. Clear sins are clear enough to judge. Hidden sins will come out. Obvious good will be praised. God will bless good thing done in secret. All we are called to do is be faithful.