V 1.
From Paul, Silvanus, and Timothy to the congregation of Thessalonians who belong to God our Father and the Lord Jesus Christ.
NIV.  Paul, Silas and Timothy.  To the church of the Thessalonians in God our Father and the Lord Jesus Christ.

This letter was from Paul, Silas and Timothy who probably set up the Thessalonian church when they visited on their travels.  Paul did not set up churches, then move on and forget about them.  He loved the people of the new churches, with all their failings and weaknesses,  and keenly felt his responsibility to continue to nurture them, even if far away - or in prison.  While Timothy travelled with him, he was training Timothy to do the same. The people of the congregation of Thessalonians, like all converts, belong to God and Paul was God's worker, sent to care for them.  He took his responsibility seriously.

New Bible Commentary, p 1278  The church was a small group of believers who met in a home or perhaps in a handful of homes.

V 2.
Grace to you and peace from God the Father and the Lord Jesus Christ.

Paul's prayer for the people he loved and nurtured: that they wouold know God's grace and the peace that only God can give.

V 3.
Our thanks are always due to God for you, brothers.  It is right that we should thank him, because your faith increases mightily, and the love you have, each for all and all for each, grows even greater.

As always, Paul gave thanks and praise to God for the faithful people in the churches he established and came into contact with. He gave thanks for the signs that the people of Thessalonica were growing in their faith.  He wanted them to know that he gave thanks for them as a means of encouraging them through some difficult times.

It's important that we encourage our fellow Christians, even when it may seem to us that they're doing OK.

V 4.
Indeed we boast about you ourselves among the congregations of God's people, because your faith remains so steadfast under all your persecutions, and all the troubles you endure.

When Paul visitied young churches, he talked about other young churches - what was happening to them and how they coped with their trials.  In this way he encouraged each congregation to know that they were not alone in persecution and that others were remaining faithful too.

Quest Study Bible note, p 1638.  It's possible that secular leaders, thinking that Christians undermined the stability of thepolitical scene, had begun to harass them.  Pockets of persecution may have broken out against a family or a leader, and perhaps these had become frequent enough totrouble the church.  Some people, no doubt, were beated, while others may have been imprisoned or robbed of their livelihood.

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V 11.
With this in mind we pray for you always, that our God may count you worthy of his calling, and mightily bring to fulfilment every good purpose and every act inspired by faith,
NLT.  And so we keep on praying for you, that our  God will make you worthy of the life to which he called you.  And we pray that God, by his power, will fulfil all your good intentions and faithful deeds.

Paul prayed for the young church experiencing persecution.  But he didn't pray that the persecution be taken away as I think I might have done!  Instead, he prayed that the people would be made worthy of what God called them to do - in spite of the persecution.  He  also prayed that God would use their lives to build his kingdom, which was what they were on about.  Paul wasn't talking to them from some comfortable office, but from his own personal experience of persecution.  They would understand  that he knew what he was talking about.

V 12.
so that the name of our Lord Jesus may be glorified in you, and you in him, according to the grace of our God and the Lord Jesus Christ.
NLT.   Then everyone will give honour to the name of our Lord Jesus because of you, and you will be honoured along with him.  This is all made possible because of the undeserved favour of our God and Lord, Jesus Christ.

This should be the primary purpose of the church - through times of both plenty and persecution - that the name of Jesus might be glorified.  We must strive to make sure that, when our church does some good things for people, it is seen to be done not because the church people are kind or loving or 'that sort of people', but in response to what Christ has done.