From Paul and Timothy to the saints, bishops, and deacons in Philippi. (1:1)
Most scholars believe that Philippians was written by Paul, but not as a single letter. It is thought to be a composite of at least two separate letters. (Wikipedia: Epistle to the Philippians)
Philippi was a city in northeastern Greece. Paul visited it on his second missionary journey (Acts 16:9-10, ca. 49-50 CE).
Timothy was a companion of Paul who was circumcised by Paul in Acts 16.
I am thinking of you while I am here in prison. (1:7)
Paul claims to be writing from prison, but, if so, no one seems to know where that might have been.
Epaphroditus is only mentioned in Philippians. He was a messenger from Philippi, who delivered some money from the Philippian church to Paul. (See 4:18)
And finally (3:1)
Finally? It seems a bit strange to say "finally" in the middle of a letter.
That's one of the reasons Bible scholars think Philippians may have been cut and pasted from several letters.
Beware of dogs (3:2)
I'm not sure who Paul is calling "dogs" here, but in the Bible, it usually refers to non-Jews, temple prostitutes, or (possibly) homosexuals. (See Dt 23:18, Mk 7:27-28, Rev 22:15)
But since, in this passage, Paul is condemning Christians who say gentile converts must be circumcised, he is probably referring to them.
Euodia and Syntyche (4:2)
Euodia (the KJV mistakenly has "Eudodias") and Syntyche were two women in Philippi who were involved in some kind of argument. It is interesting Paul permitted women to work with him in his preaching activities.
I can do all things with Christ strengthening me. (4:13)
Christians are superhuman and omnipotent. They can do whatever they want.
This is often used by Christian athletes - as made famous by Tim Tebow.
I have received from Epaphroditus the gift that you sent. (4:18)
I have received from Epaphroditus the gift that you sent.