What DOES that mean? Paul wasn't continuously saying prayers (in the conventional sense of "praying"), for example, sometimes he preached or wrote epistles. Even Jesus, teaching us how to pray, said, "WHEN you pray..." implying it wasn't the only thing we'd ever do.
Perhaps Paul means that we should always be in a prayerful attitude? Praying is a stance of the heart more than a forming of words, right? We should always be conscious of God's presence and seek to include Him in everything, always maintaining a relationship with Him, even when we are concentrating on our work or some other task, right?
That sounds good, and the concept is certainly valid. The problem is, this isn't what "prayer" usually means in Scripture. In the Bible, the term is used almost exclusively for asking God for things. If we ask for ourselves, that's "supplication," if we ask God to do something for others, that's "intercession."
True, there are other forms of prayer: confession, praise and thanksgiving; and they are generally identified by those words. But "praying" is almost always asking God for something. Sometimes we try to be more spiritual than God! It is good to always be conscious of His presence and include Him in all you do. But Paul is talking about asking. In the Bible, prayer is more tangible and specific and, dare I say, even practical, than we tend to make it.
So we have to look at the other term: "ceasing". Paul told many of the churches, "I am always praying for you." I don't think he meant that he continually said to God, "Bless everyone, bless everyone, bless everyone..."
What I think he meant was, whenever you want or need something that you don't have, the FIRST thing you should do is ask God for it (if you can't ask God for it, you shouldn't oughta be wantin' it, pardner). Whenever the needs of others come to mind (which will be frequently, in fact, almost constantly if you are truly a person of love), you should intercede for them.
It's easy to forget that God is the source of everything good that we or others ever receive, have or obtain. It's easy to justify lusting and going after lots of things that we wouldn't feel right asking God for.
Sometimes we don't go to God for the things we want because we prefer our independence.
Often we don't pray for others as we ought because we still haven't gotten very good at this whole "love" thing and are generally too obsessed with ourselves to give a prayer.
It's good to know what's wrong with us. That's the first step to becoming better people than we currently are. And it's an absolutely necessary step.
So... pray without ceasing.