This is an absolutely huge question, and one that other flaired users here can answer better than I. However, as someone with military aspirations and a strong amateur interest in the subject, I will do my best.

You are correct in that the First World War had began with a mobile phase before degenerating into trench warfare in the autumn of 1914. Very broadly speaking, and without getting into the reasons as to why they were fighting, the German Empire hoped to rapidly knock France out of the war with a massive sweep through Belgium to trap the French Army in a pocket between Paris and the Franco-German border. They would then turn their troops east to deal with France's ally Russia bearing down on East Prussia (this is commonly called the Schlieffen Plan after the German General who established the concept, though it's a misleading name as there were in fact several, continually-updated plans). The whole concept was ludicrously optimistic on the German Army's part: it involved hugely-elongated logistics trains that could not possibly support the advance, and the French and Belgian road net was not dense enough to support the number of troops needed to overcome the French Army. Thus, the First Battle of the Marne, where the Germans were stopped on the River Marne east of Paris, is arguably the first turning point of the war since it ended German hopes of a quick victory in the west and put it in a situation where it would inevitably be ground down under the superior industrial and numerical might of the Entente powers - Britain, France and Russia.

After the Marne, the Germans went on the defensive at the First Battle of the Aisne and dug in. Throughout the autumn of 1914, the Germans and French attempted to outflank each others' lines, each attack foundering amid hastily-dug trenches and the lines moving further and further north until they reached the Belgian coast. The last possibility of a German breakthrough, which would conceivably have endangered the British Expeditionary Force's supply lines if Germany could take the Channel Ports, was halted by the BEF at the First Battle of Ypres at the end of 1914.

Trench warfare was nothing new, and earlier wars like the Second Boer War and the Russo-Japanese War had proved that soldiers, properly supported by artillery, could take enemy trench lines. What was entirely different was the scale: million-man armies were crammed into the confined space between the Belgian coast in the north and the Vosges Mountains in the south, allowing them to not just hold, but continually deepen and improve a continuous front. There was no flank to turn or an indirect approach to take; every attack would have to be a frontal one. Note that the same is not necessarily true for the Eastern Front, where the vast space allowed for greater mobility, though this is beyond what I've read. The early battles of 1915 such as Neuve Chapelle, the Champagne and Artois offensives, and Festubert showed that the line would not be broken easily: attacks would have to be preceded by tremendous artillery preparation, and while the enemy first line might be taken, difficulties in communication meant that reserves could not be rapidly called forward. Meanwhile, the enemy could fall back on an intact telephone and transport net and quickly bring forward his own reserves to seal the breach.

The Entente strategy, developed at the Chantilly Conference in late 1915, envisioned multiple, simultaneous offensives on all fronts to prevent the Germans and Austro-Hungarians from using their interior lines to rapidly shift troops from one front to another and pin down their reserves. It was envisioned that this would steadily grind the German Army down and exhaust its manpower reserve as the blockade by the Royal Navy began to bite. The Entente, with its greater population, could outlast them.

This was an entirely sensible strategy, but it was completely thrown out of kilter when the Germans launched a massive assault on the French fortress-city of Verdun in February 1916: Verdun was a powerful symbol of French defiance from the days of the Revolution and could not simply be abandoned, and the Germans knew it. Whether the Germans intended to break through or just "bleed the French Army white" by forcing them to commit troops to its defence remains controversial (Mods pls halp), but the need to defend Verdun dramatically reduced the French commitment to the offensive planned on the Somme in mid-1916. Throughout the battle, the French slowly, and with agonising losses, clawed back territory by developing new artillery tactics (for example, short-but-massive "hurricane" bombardments on narrow fronts to really smash up the German defences) and a proper system of fire-and-manoeuvre for infantry.