To "pray without ceasing" basically refers to recurring prayer, not nonstop talking. Thus it is to be our way of life - we're to be continually in an attitude of prayer.
Charles Haddon Spurgeon offered this vivid picture of what praying at all times means:
"Like the old knights, always in warfare, not always on their steeds dashing forward with their lances in rest to unhorse an adversary, but always wearing their weapons where they could readily reach them, and always ready to encounter wounds or death for the sake of the cause which they championed. Those grim warriors often slept in their armor; so even when we sleep, we are still to be in the spirit of prayer, so that if perchance we wake in the night we may still be with God. Our soul, having received the divine centripetal influence which makes it seek its heavenly center, should be evermore naturally rising towards God Himself. Our heart is to be like those beacons and watchtowers which were prepared along the coast of England when the invasion of the Armada was hourly expected, not always blazing, but with the wood always dry, and the match always there, the whole pile being ready to blaze up at the appointed moment. Our souls should be in such a condition that ejaculatory prayer should be very frequent with us. No need to pause in business and leave the counter, and fall down upon the knees; the spirit should send up its silent, short, swift petitions to the throne of grace.
A Christian should carry the weapon of all prayer like a drawn sword in his hand. We should never sheathe our supplications. Never may our hearts be like an unlimbered gun, with everything to be done to it before it can thunder on the foe, but it should be like a piece of cannon, loaded and primed, only requiring the fire that it may be discharged. The soul should be not always in the exercise of prayer, but always in the energy of prayer; not always actually praying, but always intentionally praying.
I think of praying at all times as living in continual God consciousness, where everything we see and experience becomes a kind of prayer, lived in deep awareness of and surrender to our Heavenly Father. it is something I share with my Best Friend - something I instantly communicate with God. To obey this exhortation means that, when we are tempted, we hold the temptation before God and ask for His help. When we experience something good and beautiful, we immediately thank the Lord for it. When we see evil around us, we ask God to make it right and to allow us to help accomplish that, if it is according to His will. When we meet someone who does not know Christ, we pray for God to draw that person to Himself and to use us as faithful witnesses. When we encounter trouble, we turn to God as our Deliverer.
Thus life becomes a continually ascending prayer: All life's thoughts, deeds, and circumstances become opportunities to commune with our Heavenly Father. In that way, we constantly set our minds "on the things above, not on the things that are on earth" (Col. 3:2)
Since the ultimate purpose of our salvation is to glorify God and to bring us into intimate, rich fellowship with Him, failure to seek God in prayer is to deny that purpose. How would a friend feel if you never talked to them or acknowledged their presence? Yet that's how we treat the Lord when we fail to pray.
Our fellowship with God is not meant to wait until we are in heaven. God's greatest desire, and our greatest need, is to be in constant fellowship with Him now, and there is no greater expression or experience of fellowship than prayer.
~Alone with God~