A - Thanks for the question.
The marks Paul is talking about are the brands that slave owners would mark their slaves with. Since Paul has been marked by the beatings he has suffered, for the name of Jesus, he is "marked" as Christ's slave.
Paul talks about this bodily sign of Christ in other letters.
"We are afflicted in every way, but not constrained; perplexed, but not driven to despair; persecuted, but not abandoned; struck down, but not destroyed; always carrying about in the body the dying of Jesus, so that the life of Jesus may also be manifested in our body." -2 Cor 4:8-10He also talks about being Christ's slave in Romans:
"Paul, a slave of Christ Jesus." - Romans 1:1In the context of this passage we also see that Paul is talking about circumcision being a sign in the flesh of the Old Covenant, which is no longer needed. Thus, the only sign in the flesh needed now are those scars he bears for proclaiming Christ. His scars are the sign of his being Christ's slave - not his circumcision. One of the first heresies the early Church, including the apostles, had to deal with was the Judaizers. They believed you had to be circumcised, and hold the other Mosaic laws, to be a Christian. Paul was fighting against this false teaching in the letter to the Galatians.
Now, as for the stigmata - this verse is where the concept originates from, being the only time the Greek word is used in the Bible. The word "stigmata" is the plural word for "mark" or "brand".
Later, the word was used to describe, more specifically, the wounds on certain Christians that resembled Christ's wounds. St. Francis is the first recorded person with stigmata.
St. Thomas Aquinas has the following to say about this passage:
For stigmata are, strictly speaking, certain marks branded on one with a hot iron; as when a slave is marked on the face by his master, so that no one else will claim him, but quietly let him remain with the master whose marks he bears. And this is the way the Apostle says he bears the marks of the Lord, branded, as it were, as a slave of Christ; and this, because he bore the marks of Christ’s passion, suffering many tribulations in his body for Him, according to the saying of 1 Peter (2:21): “Christ suffered for us, leaving you an example, that you should follow his steps”; “Always bearing about in our body the mortification of Jesus, that the life also of Jesus may be made manifest in our mortal flesh” (2 Cor 4: 10).I hope this helps.