Question: "What does it mean to pray in Jesus' name?"
Answer: Prayer in Jesus' name is taught in John 14:13-14, "And I will do whatever you ask in my name, so that the Son may bring glory to the Father. You may ask me for anything in my name, and I will do it." Some misapply this verse, thinking that saying "in Jesus' name" at the end of a prayer results in God's always granting what is asked for. This is essentially treating the words "in Jesus' name" as a magic formula. This is absolutely unbiblical.
Praying in Jesus' name means praying with His authority and asking God the Father to act upon our prayers because we come in the name of His Son, Jesus. Praying in Jesus' name means the same thing as praying according to the will of God, "This is the confidence we have in approaching God: that if we ask anything according to his will, he hears us. And if we know that he hears us—whatever we ask—we know that we have what we asked of him" (1 John 5:14-15). Praying in Jesus' name is praying for things that will honor and glorify Jesus.
Saying "in Jesus' name" at the end of a prayer is not a magic formula. If what we ask for or say in prayer is not for God's glory and according to His will, saying "in Jesus' name" is meaningless. Genuinely praying in Jesus' name and for His glory is what is important, not attaching certain words to the end of a prayer. It is not the words in the prayer that matter, but the purpose behind the prayer. Praying for things that are in agreement with God's will is the essence of praying in Jesus' name.