Peace of Christ (15): Paul here continues the topic of Christian harmony.[1] The way we treat others is to reflect the peace we have from Christ.[2] Our inner peace is what makes us maintain peace with others. God placed every believer into a church for this very purpose ("to which you were called in one body"): that we would give to others what we received from the Lord.[3] And we do this with an attitude of gratitude.[4] We are a thankful people of peace-makers.
Word of Christ (16): We are also to let the word of Christ dwell in our church richly (not sparingly!).[5] The word of Christ is the truth about and from Christ.[6] We reinforce this message ("teach"), and we admonish one another against sin with this message. Paul highlights one specific way to accomplish this: singing (Eph. 5:19). Now, singing could degenerate into a meaningless habit. Thus, we are to sing "with thankfulness" and with sincerity ("in your hearts") to God.
Name of the Lord (17): Then lastly, before addressing the specific groups in the church (3:18ff.), Paul adds a comprehensive command to "do all" things in line with the character and authority of the Lord Jesus.[7] He should be manifest in everything we do. And we do all this with thankfulness. We keep coming to God through Christ (Heb. 10:19-22) with expressions of thanks (1 Thes. 5:18).
[1] 3:15 actually starts with "And [kai (καί)]" (ESV, NKJV). Harmony (3:12-14) is still the focus in these verses.
[2] Every believer understands this peace. It is the peace with God that he has through Jesus (1:20; Rom. 5:1).
[3] As we received forgiveness from the Lord, we extend it to others (3:13). So we do with peace. Cf. Phil. 2:1-2.
[4] "Be thankful" is a stative, describing a state of being. We have a thankful attitude as a people blessed of God.
[5] "Richly dwell within you [plural]." The one-anothers here show that the focus is on the church community.
[6] This is both the message about Christ (1:28; 1 Cor. 2:2) and the message from Christ (1 John 1:5; John 8:31).
[7] In Scripture, "in the name" refers to an action done in concert with the character of the person in question (cf. Matt. 10:41). "In the name of the Lord" specifically appeals to the Lord's authority (cf. 1 Cor. 5:4).