World War One was at the dawn of the modern military age.
Military leaders had to adapt to new technologies with new strategies.
Near the beginning of the war, soldiers would just stand up out of their trenches and shoot each other.
Later, elaborate tactics and new technologies were employed.
The battle of Vimy Ridge details the adoption of no less than six strategic firsts: creeping barrage, units of 10-15 men each with their own map (instead of 50 men with one map), suppressive gunfire, flash spotting, and others.
Here is a brief look at some of the tactics.
The tactics employed at Vimy Ridge allowed for the attacking Canadians to lose only 10k casualties while taking a heavily fortified ridge from the Germans and inflicting 30k casualties on the defenders.
Strategy played a huge role.
I highly recommend reading both of the linked articles in their entirety.
They are exciting and informative.
history.stackexchange.com/users/71/Dale
The Allies never did make a real breakthrough in the West.
At the end of the war, they were pushing the German Army back, but never breaking through.
The Germans did, against the Allies, but it's a matter of question whether German offensive practices were better than Allied, or Allied defensive practices were worse than German.
Given that the British and French were usually attacking, and that the areas the Germans hit were generally the more lightly defended, my guess is that it's a matter of worse Allied defense.
The main problems the offensive faced were communications and logistics.
Breaking through trench lines was a matter of coordinating infantry and artillery, and as the infantry advanced it lost all contact.
The infantry would hit later trench lines at more and more variable times, so the artillery couldn't coordinate.
Moreover, it was almost impossible to supply forces as they advanced; this is what caused the failure of the German Spring 1918 Offensive.
At the very end of the war, the Allies were experimenting with putting radios on tanks, to keep contact with the rear (including the artillery), and that might have made breakthroughs possible.
Tanks were also useful for fire support, although very unreliable in those days, so breakdowns were very frequent.
Defensive positions in WWII were often deeper and better held than those of WWI, and were often broken through by well-planned attacks.
Aircraft and tanks provided mobile firepower, there were good cross-country trucks, and radios allowed artillery to support the attack effectively until the limits of its range.
history.stackexchange.com/users/90/David Thornley
A multitude of approaches were taken, not just by the Allies but by the Germans as well:
Fire and move tactics (akin to modern infantry tactics).
Lighter, more portable machine guns that could be carried forward to support the attack Better artillery tactics, particularly to support the infantry attack.
Penetration tactics (on the German side in particular), attacking through the weakest points and leaving follow up units to deal with outflanked strong points.
Various technological advances (tanks, planes, gas)
Most attacks were initially successful; however, it was extremely difficult to bring up support to maintain the ground taken, and counter attacks would defeat gains made.
history.stackexchange.com/users/97/runequester
The Allied forces had more or less developed a pretty successful system of attacking by the end of the war.
It did rely on having greater material resources than the Germans.
Constant pressure and a large number of well resourced "bite and hold" attacks.
This concentrated of wearing down the German army in a pretty brutal war of attrition.
Each attack was limited, kept in range of the supporting guns, the objective was not to breakthrough but break in.
Given the limitation and problems 'exploiting' or attempted 'breakthrough' , Allied strategy was not aimed at these.
(the problems just could not be solved).
The quick succession of bite and hold attacks would eventually wear down the German reserves, and them there would be some sort of breakthrough.
history.stackexchange.com/users/3559/pugsville
The tank was one major development that helped break through the trenches.
The British Mk1 tank was employed with some degree of success at the Battle of Cambrai for example: http://en.wikipedia.org/wiki/Battle_of_Cambrai_(1917 )
The British gained more ground here in six hours than they gained in three months at Ypres for example.
However, due to lack of infantry support most of the ground gained was lost to a German counter-attack soon afterwards.
history.stackexchange.com/users/732/davidjwest
Reading the answers to  this question about WW1 Western Front , it appears that breaking through the trenches was a very difficult task to do. How did the Allies finally make it? Was it the invention of new tactics and technology? Or was it just the numerical advantage caused by the entry of the Americans?
What new technology/tactics allowed the breakthrough of the trenches in WW I western front?
history.stackexchange.com
<20th-century><world-war-one><technology><military>
history.stackexchange.com/questions/101
history.stackexchange.com/users/84/Louis Rhys
Advances in military tactics, lighter machine guns and technological advances like radios helped to break through the German defenses. The tank was one such development.  The allies were likely able to breakthrough enemy trenches as a result of more refined tactics and technological advances. Tanks will have made it easier to break through the trenches initially, but without infantry support, these gains would be temporary. Therefore, a change in infantry tactics may also have played a significant part in making these gains.
A change in tactics, such as 'bite and hold', may have played a part in allowing infantry to make greater gains in trench warfare. Advances in technology, such as tanks, will doubtlessly have significantlly contributed towards being able to take enemy trenches.