The power structure of Dune is quite convoluted as you might imagine. Ostensibly the power is divided into three parts: the Landsraad, the Emperor and the Guild. Unfortunately this is not realistic. It gives no mention of the Bene Gesserit or CHOAM who hold a very great deal of the power. Nonetheless most of the power is indeed concentrated in the three pillars of power.
The Landsraad is a confederation of planets, each owned by a house. The houses are split up into two groups: the Houses Major and Houses Minor. The only difference seems to be the amount of money and therefore power the house holds. The structure here, as in much of Dune, is feudal with each of these houses controlled by a single normally male "head of the family". The most notable are obviously Duke Leto I of House Atreides and Baron Harkonnen of House Harkonnen. The Landsraad is the oldest organisation in Dune but it changed greatly over the years that preceded the events of Dune.
It can be thought of as being in direct opposition to the Emperor. Before the Emperorship was founded the Landsraad had no armies and was mostly a decision making convocation. The rise of the Sardaukar and the Imperial House Corrino forced the Landsraad to become a military power because only through unity could the Great Houses hope to balance the massive military power the Sardukar represents.
Each of the Houses of the Landsraad maintains an army both to maintain stability in the feudal empire and to keep their planet's population in check. They also maintain a collection of atomic weapons which are not to be used against human enemies.
The Emperor or The Imperial House
The emperor is the most powerful man in the universe. He holds all the cards; he controls the huge military steam-roller that is the Sardaukar, he controls all trade though being the head of CHOAM and he is the richest because he requires tithes to be paid by each of the houses.
The Imperial House is in direct opposition to the Landsraad. Historically the Imperial House was a major house that gained too much military and financial power and forced the rest of the Landsraad to accede to its wishes. Emperorship, while it is very stable, is conferred on the most powerful house and occasionally this changes either by military conquest (once*) or by astute marriage (once*). The Emperor will tend to guard his family's holdings very jealously though plots and intrigue rather than all-out war.
Despite all his huge power the emperor is one of the less important characters in the book Dune but in later books the post gains more attention.
* This was the same time.
The Spacing Guild
The Spacing Guild holds a monopoly on space travel. It, therefore, holds the empire in a stranglehold and just allows the money and spice to flow. Although, as Frank Herbert points out, just like any good parasite, the Guild must keep its host healthy if it is to flourish. It is mostly a silent partner and only occasionally involves itself in plots directly. Its power lies entirely in its ability to effectively maroon any planet (except Arrakis) it doesn't like by not allowing wealth or anything else to flow in or out. This power has never been used to my knowledge but its threat is enough to make even the emperor think twice about crossing them.
Their only real care in the universe is maintaining their supply of spice. They require this to "fold space" and are all so hopelessly addicted to it that to be without it would mean unimaginable pain followed by death.
The Bene Gesserit
The Bene Gesserit make a point of avoiding actually holding positions of power. They are, however, drawn to it like flies. These people are the king makers of the Dune universe. They can influence whole populations as easily as they manipulate individuals and while they tend not to use their considerable power, for fear of drawing too much attention, they will use it subtly to sway opinions to their favour.
Their ostensible power is almost entirely in the fact that they are trustworthy beyond reproach. A Bene Gesserit promise will never be broken but it may not be kept in the way you expect. Their function across the empire is as advisors and truthsayers who can immediately tell if someone is lying or being underhanded.
In later Dune novels the Bene Gessert become arguably the most powerful force in the universe.
CHOAM - The Combine Honnete Ober Advancer Mercantiles
This is the trading organisation of the Dune universe. It has a hand in every pie there is from the spice melange to soostones. This is its power. The heads of great houses are appointed by the emperor to CHOAM directorships. A CHOAM directorship provides the opportunity to make huge sums by gaining preferential treatment to your goods.
CHOAM's power is a lot like the Guild's power. It has the power starve economies but at the ultimate expense of CHOAM itself. So, while it is a major player in the power game it essentially does not have the independent power of even the Bene Gesserit.
Information gained from the Dune series and The Dune Encyclopedia by Dr. Willis E. McNelly authorised by Frank Herbert.
Information not gained from the Dune Prequels!