While there are many ways that Cannons changed fort design and general defences, the two most important ways are probably this:

• The Star Fort, or Killing Zones:

While often it is easy to see the advent of the Cannon in European Warfare as simply being "Cannon blows hole in wall: Wall becomes irrelevant" once defence engineers adapted to this new problem, they actually used gunpowder quite effectively, integrating it into the design of fortifications. The most famous example of this is the Star Fort which was prominent in the 18th century, but even before then it's basic principles were being used to stall attackers. In the 16th Century, instead of simply firing out of the castle straight at attackers from Square Keeps and Curtain Walls, Cannons integrated into the walls would fire almost parallel to the direction of the walls, in new Diamond shaped forts. This would make it impossible for any attackers to reach the walls without being exposed to several points at which a cannon could hit them. This essentially created Killing Fields at the foot of walls, making storming Castles much more costly. Examples of Forts like this would be the South Coast forts of Henry VIII, which ultimately were unused but are very impressive nonetheless, like Hurst Castle (which is an odd-case as it is a seafort, but the principles are largely the same).

• The Glacis

The Glacis was actually invented before the advent of gunpowder: it's arguable that the various banks in Iron Age Hillforts are primitive glacis. But what is a Glacis? Well, whereas most walls are high, vertical structures designed to keep the enemy out, the Glacis was a gently sloping earth bank in front of a wall, designed to expose attackers to defenders fire from the walls, make walls stronger against undermining, and crucially to deflect missiles thrown at the walls. After the advent of Gunpowder, this became even more crucial, as knocking down the wall with missiles became a viable alternative to escalade (climbing the wall) or siege (starving them out). A Good Glacis would be constructed at such an angle that Canon shots would bounce off of them, severely delaying attempts to breach the walls. Sometimes it would also create a trench between the wall and the glacis, perfect for focused fire from defenders.

An example of the Glacis at work could be the Sieges of Badajoz in the Peninsular War: in 1811, 4000 French Garrison Troops held off 20,000 British and Portuguese Troops who were besieging them. After 2 months of constant artillery barrage, no breach had been made, and The Duke of Wellington was forced to lift the siege as a Relief Force was approaching. A year later, Wellesley besieged the Fortress City again, this time with 27000 men and 52 Siege Guns (some of the British and Portuguese Canon from the first siege had actually been destroyed by French Counter-Battery fire from the walls: the French could fire over their own Glacis, while British fire was ineffective). Eventually the Fortress fell after one of the Outer forts that blocked a gap in the Glacis (San Cristobal) was stormed by the British, allowing the Siege Guns to penetrate the gap in the Glacis, and breach the main walls. But in Storming the Breach, the British lost 4800 men out of 27000 in a few hours, facing only 4800 starving French defenders. This is because right up until the point the British were in the Breach, French cannon could sweep them with Canister Shot, inflicting terrible casualties. In fact, no British troops actually made it through the breach: the fortress was taken by an escalade on the older Castle, which was built at the top of a cliff, that had actually been intended more as a distracting action to pull defenders away from the breach. No artillery bombardment of this section had been made, and the audacity of the British climbing a cliff and then scaling a castle wall with ladders they had carried up stunned the French. This shows that even one of the periods most able commanders still found surmounting City Defenses incredibly difficult, despite his use of modern canons, actually being forced to resort to Medieval tactics. This would be repeated at the incredibly costly siege of San Sebastian in 1813, where Wellington actually came close to calling off the attack as he was taking unacceptable losses, only succeeding when he fired at his own troops to clear the breach of French defenders, as well as his own attackers.

Medieval castle walls could be thick, but were often many times taller than they were wide, and constructed of mortar and stone. Cannons had little trouble reducing such structures. Fortress design came to favor lower walls, backed by earth and gravel, many times wider than they were tall.

Also, fortifications featured armaments like gun emplacements for defensive use. Many forts designed post-cannon made use of geometric principles to compliment and optimize gun placement. Whereas the walls of a medieval castle were the primary defensive feature, in a post cannon fortress, the compliment and placement of defensive batteries were the primary defensive feature.

Building styles like the Star Fort became a common solution to the introduction of gun power of the battlefield in general, and cannons in particular. They allowed defenders to maximize the impact and versatility of their own gunpowder weapons while deadening that of their foes.

Other users have already explained the basics of Star Forts, but I thought I'd chip in using a source that explicity addresses the period in which gunpowder weapons truly came into their own in sieges in Europe. My source for the following answer is "The military revolution in Sixteenth Century Europe" by David Eltis, which has an chapter devoted to how siege warfare changed in the 16th century due to the increasing number and sophistication of gunpowder weapons, and the armies to use them.

Existing defences in Europe at the start of the century (tall, thin walls) were both vulnerable to artillery, as they offered a large target, and difficult to use artillery with as weapons on the tops of the walls would lead to blind spots at the base of the walls. It was the angle bastion that solved these problems; low, thick walls with an earthen bank (glacis) opposite- this forms a ditch with the fortification wall making up one side and the earthen bank the other, so the attacker could not fire his artillery at the bulk of the wall, but the defender with guns atop the wall could fire at attackers approaching up the glacis. At the corners of the fortification, a bastion would protrude at an angle (hence the name) allowing guns mounted atop the bastion to take any attackers in the ditch in a cross-fire. (Remember at this point the defenders could usually mount larger cannon that the attacker, as the defender does not to move the cannon any great distance.) The defender could fire scattershot at the attackers, while the attackers could only rely on solid shot as their guns would be further from the walls. Here is an excellent diagram from the book that compared pre and post gunpowder defences.

These defences could be built from stone at great cost, or more cheaply from mounds of earth, although unshored earthen defences would weather quickly (Eltis says "a few years").

Breaching these defences was possible with a sizeable siege train, at least 20 cannon and preferably a lot more, but would require immense logistical effort (Eltis quotes authors of millitary treatises arguing it would take from 7000-16000 cannon balls to create a single breach that was 10-12 men wide.)

Once the breach was open, the defender could then focus his weapons on the breach, achieving a concentration of fire at the breach where "Both the arquebusiers and artillery were hidden from view until the attacker stepped into the breach. At such close range every musket shot told and all defensive armour was useless." This would be done by the defender digging trenches in a crescent (referred to as a half-moon at the time) around the breach. Eltis notes that this was a terror to any attacker, and a reason for why so many assaults failed in the sixteenth century. Of course, this would require open ground behind the wall where this could be made, which was itself an important consideration when building defences.

In summary, gunpowder weapons changed the nature of assaults on fortifications by greatly improving the firepower available to defenders, through handguns and through artillery mounted on walls, but a determined and well provisioned attacker could, with sufficient time, overcome these obstacles. I'd be happy to refer to Eltis for any follow up questions. Thanks for reading!

Out of interest /u/jofwu , what fantasy series have you been reading? My own interest started similarly; reading The Wheel of Time series.