V 1.
First of all, then, I urge that petitions, prayers, intercessions, and thanksgivings be offered for all men;
NLT. I urge you, first of all, to pray for all people.  As you make your requests, plead for God's mercy upon them and give thanks.

Paul stresses that prayer is important.  He mentions all kinds of prayer.  We need to pray in different ways for different situations: each kind of prayer is equally important.

Petition: asking God for something we need or want.  Philippians 4: 19 tells us that 'this same God....will supply all your needs from his glorious riches, which have been given to us in Christ Jesus.' (NLT)  And my God will meet all your needs according to his glorious riches in Christ Jesus (NIV)  Matthew 7: 7 tells  us Ask and it will be given unto you.
(Of course, if we love God and are his obedient children, we will ony ask for things we know he wants too. God will not give us anything that goes against his perfect will or anything which harms us or anyone else.)
Philippians 4: 6 tells us Do not be anxous about anything, but in everything, by prayer and petition, with thanksgiving, present your requests to God.

Intercession.  It is important that we pray for other people.  That is part of loving others.  Paul prayed always for those in the churches he had visited.  His letters often begin with letting people know he prays for them.  He usually prayed for them to become more understanding of God's will, for them to be more loving, for them to be obedient Christians - for them to grow in their faith.

Thanksgiving.  Paul always gave thanks to God for the people he had brought to faith in Christ.  He gave thanks for their joy, for their faithfulness, for their love for each other.  He thanked God for the privilege of being his Apostle, for being his Lord.  We have so much to thank God for - this should be an easy type of prayer for us.

Paul tells us that we must pray for all people - not just those we like or love or those we thing are worthy of it.  All people means those who are not doing the right thing in their lives as well as those who are.  God has no favourites.  He wants us to have no favourites too.

V 2.
for sovereigns and all in high office, that we may lead a tranquil and quiet life in full observance of religion and high standards of morality.
NLT.  Pray this way for kings and all others in authority, so that we can live in peace and quietness, in godliness and dignity.

It is very important that we continually pray for the leaders of our country, our local area, our churches and any other people in leadership positions in our community, because their influence on the way we and others live is so vital.

New Bible Commentary, p 1297. What is significant is that Paul makes no distinction between those rulers that are just and those that are not.  He sees it as a Christian duty to pray for those whose actions affect every citizen.  But the purpose of teh prayer is that Christians may live peaceful and quiet lives in all godliness and holiness (NIV)

The way we pray for unjust rulers will be different from the way we pray for just rulers.  But we must pray for both.

V 3, 4.
Such prayer is right, and approved by God our Saviour whose will it is that all men should find salvation and come to know the truth.

Paul states that it is God's will that we pray in all these different ways for everyone.  And here's the clue about what is the most important thing for us to pray for - that everyone in the world will come to know God and accept Christ as their personal Lord.  God wants everyone in the whole world to be saved - he wants to have a close personal relationship with every human being.  Of course, because he has given us all free will and the ability to choose, there are some people who will choose not to accept Christ and not to have a personal relationship with God.  But that doesn't mean we shouldn't continue to pray for them.

V 5, 6.
For there is one God, and also one mediator between God and men, Christ Jesus, himself man, who sacrificed himself to win freedom for all mankind, so providing, at the fitting time, proof of the divine purpose;

While we are asked to pray for each other, each one of us has direct access to God, because of what Christ did for us.  Because of his taking all our sins onto his own shoulders and paying the penalty on our behalf, we have the right to go directly to God with our prayers.  We do not have to go through a priest, a saint or any other third party.  That doesn't mean we shouldn't ask other people to pray for us, but we need to remember that we can pray for ourselves (as well as for others)

We know that God understands us when we pray to him, because Jesus was a normal human being and understands how we think and feel.

If we ever have any doubts about how much God wants to be reconciled with human beings, we only have to consider what Jesus willingly went through to make that possible.

V 7.
of this I was appointed herald and apostle (this is no lie, but the truth), to instruct the nations in the true faith.

Paul continually reminds the people he writes to that he has been appointed by Christ to bring the message of the Kingdom of God to all people.  He speaks with authority - given by Christ himself.  Each Christian today has been appointed by God too, to spread the word.  Jesus gave us the commission Go therefore and make all nations my disciples.... (Matthew 28: 19)