Could someone give me some examples of precisely how one breaks the 4th commandment, or elaborate on it a little?

Thanks.

I have been told that the commandments are given in order of importance, why is it that "honour thy father and thy mother" is more important than "thou shalt not murder"?

Agree.  I've never heard that the commandments were given in order of importance but it makes sense.  The first 3 commandments are directed towards God and spiritual things.  The last 7 towards your neighbor and temporal things.  Since the 4th commandment concerns itself with obedience to temporal authority (i.e. parents, country, Church), then it stands to reason that this is of the highest importance to the operation of society.  If you don't have order, authority and hierarchy, you have chaos.  Killing someone is temporary choas, which only affects society in the short-term, while disobedience and mutiny leads to rioting, revolts and civil unrest - a long-term destruction.  

2198 This commandment is expressed in positive terms of duties to be fulfilled. It introduces the subsequent commandments which are concerned with particular respect for life, marriage, earthly goods, and speech. It constitutes one of the foundations of the social doctrine of the Church.

For the Lord honored the father above the children, and he confirmed the right of the mother over her sons. Whoever honors his father atones for sins, and whoever glorifies his mother is like one who lays up treasure. Whoever honors his father will be gladdened by his own children, and when he prays he will be heard. Whoever glorifies his father will have long life, and whoever obeys the Lord will refresh his mother.²⁴
O son, help your father in his old age, and do not grieve him as long as he lives; even if he is lacking in understanding, show forbearance; in all your strength do not despise him. . . . Whoever forsakes his father is like a blasphemer, and whoever angers his mother is cursed by the Lord.^25.

Whoever told you that is wrong:

http://www.vatican.va/archive/ccc_css/archive/catechism/p3s2.htm

The unity of the Decalogue
2069 The Decalogue forms a coherent whole. Each "word" refers to each of the others and to all of them; they reciprocally condition one another. The two tables shed light on one another; they form an organic unity. To transgress one commandment is to infringe all the others.30 One cannot honor another person without blessing God his Creator. One cannot adore God without loving all men, his creatures. The Decalogue brings man's religious and social life into unity.


St. James chapter 2
http://www.drbo.org/chapter/66002.htm

"And whosoever shall keep the whole law, but offend in one point, is become guilty of all."

Relative Importance Of The Preceding And The Following Commandments

The preceding Commandments are supreme both in dignity and in importance; but those which follow rank next in order because of their necessity. For the first three tend directly to God; while the object of the others is the charity we owe to our neighbour, although even these are ultimately referred to God, since we love our neighbour on account of God, our last end. Hence Christ our Lord has declared that the two Commandments which inculcate the love of God and of our neighbour are like unto each other.

Importance Of Instruction On The Fourth Commandment

The advantages arising from the present subject can scarcely be expressed in words; for not only does it bring with it its own fruit, and that in the richest abundance and of superior excellence, but it also affords a test of our obedience to and observance of the first Commandment. He that loveth not his brother whom he seeth, says St. John, how can he love God whom he seeth not? In like manner, if we do not honour and reverence our parents whom we ought to love next to God and whom we continually see, how can we honour or reverence God, the supreme and best of parents, whom we see not? Hence we can easily perceive the similarity between these two Commandments.
The application of this Commandment is of very great extent. Besides our natural parents, there are many others whose power, rank, usefulness, exalted functions or office, entitle them to parental honour.
Furthermore.(this Commandment) lightens the labor of parents and superiors; for their chief care is that those under them should live according to virtue and the divine Law. Now the performance of this duty will be considerably facilitated, if it be known by all that highest honour to parents is an obligation, sanctioned and commanded by God ...

                                                                              Why Parents Should Be Honoured
In his instructions the pastor should chiefly emphasise the obligation of honouring all who are entitled to be called fathers, especially our natural fathers, of whom the divine Commandment particularly speaks. They are, so to say, images of the immortal God. In them we behold a picture of our own origin; from them we have received existence, them God made use of to infuse into us a soul and reason, by them we were led to the Sacraments, instructed in our religion, schooled in right conduct and holiness, and trained in civil and human knowledge.

"And Thy Mother"

The pastor should teach that the name mother is mentioned in this Commandment, in order to remind us of her benefits and claims in our regard, of the care and solicitude with which she bore us, and of the pain and labor with which she gave us birth and brought us up.

Manner Of Honouring Parents

The honour which children are commanded to pay to their parents should be the spontaneous offering of sincere and dutiful love. This is nothing more than their due, since for love of us, they shrink from no labor, no exertion, no danger. Their highest pleasure it is to fed that they are loved by their children, the dearest objects of their affection. Joseph, when he enjoyed in Egypt the highest station and the most ample power after the king himself, received with honour his father, who had come into Egypt. Solomon rose to meet his mother as she approached; and having paid her respect, placed her on a royal throne on his right hand.
We also owe to our parents other duties of respect, such as to supplicate God in their behalf, that they may lead prosperous and happy lives, beloved and esteemed by all who know them, and most pleasing in the sight of God and of the Saints in heaven.
We also honour them by submission to their wishes and inclinations. My son, says Solomon, hear the instruct­on of thy father, and forsake not the law of thy mother; that grace may be added to thy head, and a chain of gold to thy neck. Of the same kind are the exhortations of St. Paul. Children, he says, obey your parents in the Lord, for this is just; and also, children, obey your parents in all things, for this is well­pleasing to the Lord. (This doctrine) is confirmed by the example of the holiest men. Isaac, when bound for sacrifice by his father, meekly and uncomplainingly obeyed; and the Rechabites, not to depart from the counsel of their father, always abstained from wine.
We also honour our parents by the imitation of their good example; for, to seek to resemble closely anyone is the highest mark of esteem towards him. We also honour them when we not only ask, but follow their advice.
Again we honour our parents when we relieve their necessities, supplying them with necessary food and clothing according to these words of Christ, who, when reproving the impiety of the Pharisees, said: Why do you also transgress the commandments of God because of your traditions? For God said: "Honour thy father and thy mother," and "He that shall curse father or mother let him die the death." But you say: "Whosoever shall say to his father or mother, The gift whatsoever proceedeth from me, shall profit thee." And he shall not honour his father or his mother; and you have made void the commandment of God for your tradition.
But if at all times it is our duty to honour our parents, this duty becomes still more imperative when they are visited by severe illness. We should then see to it that they do not neglect confession and the other Sacraments which every Christian should receive at the approach of death. We should also see that pious and religious persons visit them frequently to strengthen their weakness, assist them by their counsel, and animate them to the hope of immortality, that having risen above the concerns of this world, they may fix their thoughts entirely on God. Thus blessed with the sublime virtues of faith, hope and charity, and fortified by the helps. of religion, they will not only look at death without fear, since it is necessary, but will even welcome it, as it hastens their entrance into eternity.
Finally, we honour our parents, even after their death, by attending their funerals, procuring for them suitable obsequies and burial, having due suffrages and anniversary Masses offered for them, and faithfully executing their last wills.

Manner Of Honouring Other Superiors

We are bound to honour not only our natural parents, but also others who are called fathers, such as Bishops and priests, kings, princes and magistrates, tutors, guardians and masters, teachers, aged persons and the like, all of whom are entitled, some in a greater, some in a less degree, to share our love, our obedience, and our assistance./

Right, perhaps "important" was an ambiguous word for me to use. I was implying that the commandments are given in order of gravity.