Prayer is a universal ministry for all Christians (Psalm 65:2; Luke 18:1-8). You don't have to have special spiritual gifts to pray, but you do have to develop a heart for it. The ministry of prayer requires intimacy with God: a sharing of His heart, mind and Spirit. In I Timothy 2:8 it is implied that you must separate yourself to it (see II Timothy 2:19-22; Psalm 66:18; I Peter 3:7). There are different aspects of prayer. That's why in I Timothy 2:1 there are four terms to describe the ministry of prayer:

Supplications = petitions, entreaties; asking for help with a need (Strong #1162 cp. #1189).
Prayers = communications with God (Strong #4335 cp. #4336).
Intercessions = word used to describe approaching a King on behalf of others or because of others. You can petition God "for" or "against" other people. See Esther 7:1-6;
Acts 4:23-30 (Strong #1783 cp. #1793).
Giving Thanks = grateful language to God, as an act of worship. Literally, "to speak well of God" (Strong #2169 cp. #2168).

Now, let's focus on Ephesians 6:18-20

Ephesians 6:14-17 tells us to stand and to have the girdle/belt of truth, the breastplate of righteousness, the shoes of the Gospel and to take the shield of faith, the helmet of salvation and the Sword of the Spirit/The Word of God. The next verse tells us that while we are doing this we are to be praying, at all times (seasons, occasions). We are called upon to pray and to supplicate (petition i.e. thinking of others).

We must pray "in the Spirit" (i.e. directed by the Scriptures and with the mind and heart of God). See Ephesians 1:15-23; 3:14-19; Acts 4:23-31 cp. Psalm 2). Jude 20-23 shows intercession and intervention working together: Build yourselves up in your most holy faith (The Word/The Gospel); praying in the Holy Spirit; caring for the lives/destinies of others.

As we pray we are to watch with all perseverance and supplications. "Watch" means "to keep alert." See the example of Nehemiah, rebuilding the walls of Jerusalem, with enemies all around (Nehemiah 4:9 cp Mark 13:32-37; 14:32-38; I Peter 5:7-9).

Pray for those in authority. - I Timothy 2:1-3
Pray for all saints. - Ephesians 6:18
Pray for God's servants who are on the front-line. - Ephesians 6:19-20 cp. Romans 15:30

o Utterance = opportunities to speak
o Boldness = without fear
o Gospel = the power of God (Romans 1:16)