The answer to this question varies greatly depending on which European power we discuss. The relationship between Great Britain and the colonies changed significantly. Prior to the war, British authorities practiced a rather hands-off approach to governing their North American colonies. As long as the colonists remained loyal and did not cause too many problems, the King and Parliament were content to let them do pretty much as they pleased. This is often referred to as a policy known as Salutary Neglect.

After the war, Britain took a more controlling approach. The war had a hefty price-tag for Great Britain, and they felt that the Colonies should help pay for it since it was essentially fought for their protection. This led to the levying of new taxes and the enforcement of commercial regulations. This more heavy-handed approach to governing the colonies would breed the resentment that would eventually lead to the American Revolution.

For France, the change in relations with their North American colonies was starker. They lost them. A condition of the 1763 Treaty of Paris entailed the ceding of all French territory east of the Mississippi River. Many French colonists returned to France. Others left for French-controlled islands in the Caribbean. Some stayed behind and became English subjects.

We must also talk about Spain. Near the end of the war, the Spanish allied themselves with the French. After the war, the Spanish lost dominion over Florida to Great Britain. There were very few Spanish colonists there at the time, and most left when Florida was ceded to the British. Little changed, however, in the Spanish territories west of the Rocky Mountains, as these remained under Spanish control.

In an interesting twist, Spain was given the French territory of Louisiana as a condition of the Treaty of Paris. Upset at the transfer of power to Spain, some of the French colonists left for the Caribbean colonies, yet others stayed to remain under Spanish rule. Some French fur traders remained further inland, particularly in the St. Louis area.

The Spanish made some attempts to establish their own populations in the territory. A meager population of Spaniards settled in New Orleans where they intermarried with the French and Cajuns there. The relationship between the French colonists and Spanish officials was always a tenuous one. In order to appease the French population, the Spanish took a light approach, allowing the continuation of the French language and customs in the territory. They did, however, impose certain trade restrictions that favored Spain. French resentment grew large enough that they drove out the Spanish authorities in October 1768. Faced with open revolt, the Spanish responded in force and violently put down the insurrection.

The Seven Years' War resulted in major changes regarding the North American colonies held by Britain and France. Britain won the war against the French and their Native American allies, but the ultimate result was more complicated than a simple victory.

The French lost control of their colonial territories in North America. They lost their grip on the Ohio River Valley west of the Appalachian Mountains, and they were pushed out of Canada. Britain expanded their control over North America by gaining the land west of the Appalachian Mountains and East of the Mississippi River, as well as territory in Canada and Florida. This proved to be a loss for the Native Americans west of the Appalachian Mountains, as they had forged more positive relations with the French and held more hostile relations with the British.

Things were not perfect for the British, however. Despite their gained territory, problems arose with the 13 colonies that would eventually form the United States. The British, who found themselves in debt after the conflict, turned to the colonists to help repay it. They believed that increasing taxes on the colonists would prove an effective way to help repay their war debt. This led to tension with and anger from the colonists, who lacked representation in British Parliament (which later became a major motivation and rallying point for independence). Furthermore, the colonists were displeased with the issuing of the Proclamation Line of 1763, which forbade colonists from settling west of the Appalachian Mountains. Many of the colonists who had supported the British in the war hoped that a British victory would wrest control of the Ohio Valley from the French and thus open it to colonial settlement. This also proved to be a point of contention giving rise to the American Revolution.

The Seven Years' War, also called the French and Indian War in North America (1756-1763), was a worldwide conflict. In North American, the conflict involved fighting for control of the Mississippi River basin and Ohio River Valley between the French and British. The British victory meant that the colonies took over this land running throughout the Ohio River Valley and southward, to the east of the Mississippi River.

After the war, the American colonists wanted to move into the land that the British had conquered from the French, but the English throne, wary of causing conflicts with Native Americans in the area (such as Pontiac, chief of the Ottawas), prohibited colonists from entering the territory that was west of the Appalachian Divide. The result was that colonists were angered by this proclamation and the limit on their freedoms. In addition, the English throne, in debt because of the war, re-instituted the Navigation Acts after a long period of what was called salutary neglect. The colonists, long used to non-infereference in their trade and economic affairs, reacted by beginning the process of resistance that would lead to the Revolutionary War. 

This question should more properly ask about the relations between England and the North American colonies.  After this war, France no longer had any North American colonies.  Therefore, the relationship was really just between England and the colonies. 

The relationship between England and the colonies was fundamentally changed by the war.  It is fair to say that the war was the factor that changed the relationship and led to the American Revolution.  This is because the war caused England to need to take closer control over the colonies.

After the war, England’s finances were in poor shape.  The war had been expensive and there were debts that needed to be paid.  Therefore, the English government did two main things.  First, it tightened its enforcement of laws (particularly laws having to do with trade and smuggling) in the American colonies.  Second, it imposed taxes on the colonies to help defray the costs of the war.  These actions angered the colonists greatly.  They had become used to being left alone in the past few decades and did not like having the government exerting control over them in new ways. 

Thus, the war severely weakened the relationship between the British government and the colonies.