The BBC has quite a good critique on the subject. It is of course a convenient way of getting good quality ash, but there is a theological significance.
The Palm Sunday palms symbolise Christ's triumphant entry into Jerusalem, and converting them into ash is a reminder that defeat and crucifixion followed soon after.
The ashing is a mark of penitence, and using the palms is also a reminder that the same people who cried "Hosanna" were before long baying "Crucify!" just as the Palm Sunday liturgy recreates. It is a mark of our own penitence for our own concupiscence (Christ's sacrifice of himself was a propitiation once for all time, including for us in our time).
Ash on its own doesn't make much of a mark, and some kind of binding agent is needed to allow them to make their statement. The ashes may simply be mixed with water as they are blessed (which is usual in most situations), but they may also be mixed with oil as a reminder of the chrismation and signing with the cross at Baptism.
In the Palm Sunday liturgy, palms are blessed outside the church and a short gospel is read about the finding of the colt and the entry into Jerusalem. The people follow, rejoicing as on the first Palm Sunday. Once inside the church the mood of the liturgy changes abruptly, reaching its climax in the Passion Gospel. Generally this is semi-dramatised and the people have their own part to play again, shouting "Crucify! Crucify!" at that point in the narrative.
It's common in Episcopalian and other high-church circles; and universal in the Roman Catholic Church.