Yes and no.
I say yes because the Somme has left a deep scar on the British psyche, and even in Canada and the United States, it colors notions of what the First World War on the Western Front was like: men being led "over the top" across a scarred battlefield and being senselessly cut down by German artillery and machine-gun fire.
This vision of the war really sunk in during the 1930s, largely as a result of the work of B.H. Liddell Hart. In the 1920s, historiography of the war was still mostly patriotic, celebrating the victory and the feats of the British army. But Liddell Hart attacked the army's upper command as being incompetent and numb to the casualties of the war. To him, the nearly 60,000 casualties, including almost 20,000 dead on July 1, 1916, the first day of the Somme, was all the proof he needed. This was the birth of the "lions led by donkeys" myth. (That term was actually coined in the 1960s, but the fact that the idea of army incompetence lasted that long is a testament to its resiliency.)
Even today, when the general public, particularly in the UK, pictures the war, people draw on the imagery of the first day of the Somme shaped by decades of both popular and academic historians pointing to that moment as evidence of the army leadership's gross incompetence.
However, I also say no because the nature of the fighting on the Western Front was always changing. The type of fighting we see on the Somme, and the casualties that went along with it, were the product of certain circumstances and, in many ways, were not representative of how the British army fought the duration of the war in the West. Thus, we can't say, for any battle, "This is how the war was fought." Even the engagements that made up what's referred to as "The Battle of the Somme" changed as the British learned what worked and what didn't and as they altered their approach to compensate.
I won't get into the origins of static trench warfare here, but a flankless war of position presented a new problem for military thinkers. For the British army, prewar tactics emphasized fire and movement: engaging the enemy in a firefight, probing for week points, and, if possible, maneuvering around a flank and hitting the enemy from the side. With no more flanks on the Western Front as of October 1914, you had to punch through the enemy's positions instead of going around them. Once the enemy dug in and reinforced its positions with increasingly elaborate field fortifications, doing so was extremely difficult.
The British army's first attempt to overcome static trench warfare emphasized breaking through the German positions. In early 1915, Sir John French, commander of the British army at the time, delegated the planning of an assault on Neuve Chapelle to Sir Douglas Haig, commander of First Army. The plan was to break through the German position and exploit the breach, thereby rupturing the line and collapsing the front. The attack was initially supposed to be in conjunction with the French, but they backed out at the last minute. The British went ahead with the attack, though, and it was quite successful. A short artillery barrage broke the German lines and the assault troops captured their positions. However, Haig's IV Corps, under the command of Sir Henry Rawlinson, failed to exploit the breach.
This idea of a breakthrough was key to the planning of the Somme. Neuve Chapelle was a sort of proof of concept: trench lines could be broken and enemy lines, if the reserves were ready to move into the breach, could be exploited. This same general approach was adopted for the beginning of the Somme battle in 1916. That is, destroy the enemy's positions with a long, comprehensive preparatory artillery barrage, break through with infantry, and exploit the breach with mobile cavalry.
Now I'm going to have to gloss over a lot of finer details here, but suffice it to say, the breakthrough approach didn't work as well on the Somme as it did at Neuve Chapelle for a variety of reasons. The British attacked across much of their frontage. Thus, they were stretched thin in terms of both manpower and guns, so the strength of the attack was insufficient. Moreover, most of the British army comprised raw volunteers with insufficient training. Finally, after more than a year of preparation since Neuve Chapelle, the German defenses were far more complex: two highly fortified lines and a third under construction (which the British didn't even know about on July 1). The result was the single bloodiest day in the history of the British army.
By July 14, two weeks after the first attack on the Somme, the nature of the fighting had already changed. A new attack by Rawlinson applied the lessons of the earlier fighting and concentrated more men and more guns on a smaller section of the front. It managed to capture the German second line. Tanks also appeared for the first time during the latter stages of the Somme battle. The rolling barrage, the practice of walking (or creeping) your artillery fire forward as your infantry advanced behind it, was also pioneered at the Somme.
I'm again glossing here, but the fighting in 1917 differed yet again. The Germans, in response to the British "big push" on the Somme, started adopting defense-in-depth. These tactics emphasized drawing the enemy into a deep killing zone and strongly counterattacking instead of reinforcing and holding the front line. For the attacker, the hope of outright breaking through was now complete fantasy. Successfully attacking German positions thus required more guns and more troops concentrated on smaller sections of front with more moderate objectives. The new idea was to take a piece of the German line and defend it against the inevitable counterattack--in other words, to "bite and hold" a piece of the front rather than break through it completely.
I should also briefly touch upon the mobile and semi-mobile phases of fighting at the beginning and end of the war, respectively. These differed markedly from the Somme in almost every way. In August 1914, the Germans were attacking French and Belgian fortresses and advancing rapidly toward Paris (though not as rapidly as Schlieffen planned). In September, the Battle of the Marne was won by a flanking maneuver and the opening of a critical gap in the German line. At First Ypres in October, the British and German armies collided head on over open ground before taking to the spade. Later, in March 1918, the Germans busted the front open and advanced many miles over the course of a few days. The British pushed back in August at Amiens, a battle that, with its brief artillery barrage and tanks, looked nothing like the first day of the Somme.
So you can see with this very broad overview of the British army's fighting styles on the Western Front that what we think of as "the Somme" was one iteration of combat. More specifically, it was the product of thinking about breakthroughs, which themselves had their origins in 1915. There wasn't even a single type of fighting during the Somme battle. The offensive transitioned, over the course of a few months, from a grand breakthrough to more limited attacks utilizing rolling barrages and early bite and hold tactics. Once the Germans adopted defense-in-depth, large, front-spanning offensives akin to the early Somme gave way to smaller, more concentrated attacks aimed to chip away at the German line.
To conclude, the first stages of the Somme battle are indeed "representative" of the First World War in collective consciousness, particularly in the United Kingdom, but also to a lesser extent in Canada and the United States. In terms of the nature of fighting, however, the Somme can't be regarded as typical in any sense: the tactics used in early July 1916 were the products of early concerted attempts to overcome the realities positional warfare. No single type of fighting even characterized the Somme. Lessons were learned, new tactics were employed, and the character of the fighting changed.
Sources: Herwig, The Marne; Strachan, The First World War; Sheffield, The Chief; Beckett, Bowman, and Connelly, The British Army and the First World War; Wilson and Prior, Command and Control; Griffith, Battle Tactics; Cook, Shock Troops; War Office, Field Service Regulations 1909; Terraine, The First World War; Liddell Hart, The Real War
Could you clarify what you mean by "representative"?