• What drove the fighting back during the Era of the Waring States was basically the 'cycle of hatred', as they usually refer to in the Naruto universe: the killing of some loved one or someone from the clan could not go unpunished, and so war and death were always around. This cycle was started, according to the Wiki page, as each nation "crusaded for more rights and land". Back then, the nations' shinobi were not yet organized into villages, so any existing Feudal Lord (which I am not sure existed) would not have power over any clan, so he could not force them to make a truce. Clans responded to Feudal Lords as long as they got paid, meaning that there wasn't even any particular affiliation to the nation itself. Money, lands and rights initially spawned the conflicts, and then the 'cycle of hatred' settled in.
  The ending of this Era is marked by the founding of Konoha by the Uchiha and Senju clans. The heads of each of these clans shared the vision that peace was possible, as long as one could reach out to the other. The example of Konoha was followed in other countries, thus creating the Five Great Shinobi Countries.
  As a means to help settle the peace between these five new-born villages, Hashirama Senju (then First Hokage), who was able to control Bijuu with his Wood Release techniques, distributed said beasts between the other countries. This was done with the intention of distributing the power equally between the countries. However, at least according to this page, this may have actually have increased hostilities. This may have been due to the First Hokage's death, which made Bijuu control harder, and which in turn lead to the 'storing' of Bijuu inside people (Jinchuuriki).
• The above seems to be the reason for the start of the First Shinobi War, meaning that the fighting was due to rights (meaning some states/villages were displeased with the power distribution) or to the already said 'cycle of hatred' (meaning villagers and Kage were displeased by the fact that their people had been lost, possibly because of Bijuu going berserk or having to be stored inside loved ones). This war was settled with a peace treaty, but not before all five countries had suffered serious damage. During this war the Second Hokage (Tobirama Senju) died, but appointed Hiruzen Sarutobi as Third Hokage first.
• After about twenty years after the peace treaty, the economic disparity between countries was a serious problem, and the countries started to use their military forces to expand their territories under the pretext of expending fair rights. This lead to the start of the Second Shinobi War, which took place mostly in smaller countries like Amegakure, leaving the main countries mostly unarmed. This was the war in which Jiraiya, Tsunade and Orochimaru fought. This was also the war that marked the foundation of Akatsuki, since it left Nagato, Yahiko and Konan orphans, and most of the bloodshed took place in their own country. Although the exact events that led to the start of the war are uncertain, Nagato said that the war was started by Konoha.¹ What settled this war is not stated.
• The Third Shinobi War was caused by a decline in the power of the Five Great Nations. This led to continuous fights along the borders with smaller nations, which in turn led to a war that ended up involving all Five Great Nations. This was the hardest of the first three wars, since all five nations were faced with war shortage. This was began to turn in Konoha's favour after the incident in Kannabi Bridge (shown in the Kakashi Gaiden). In this war Sasori gained reputation and made a name for himself, Minato Namikaze fought with the Fourth Raikage and Killer B, and it was the war in which Kakashi and Obito fought. This was also the war that lead to the dying of Yahiko, leading to a change in the direction of Akatsuki. How this war was settled is not stated.
• The Fourth Shinobi War was caused by the Five Kage's refusal to surrender the Bijuu that remained to be captured (Hachibi and Kyuubi) over to the Akatsuki, leading to a declaration of war from Tobi's part. Tobi's need for the Bijuu rests in his Eye of the Moon plan, which intends to cast the whole world into a genjutsu, thus bringing the illusion of peace. As the Five Kage refused to give in to this plan, a declaration of war was made. Faced with this, the Five Kage and Land of Iron's Samurai leader Mifune agreed to fight side by side, thus creating the first ever Shinobi Alliance. This war is still ongoing.

As a side note, it is important to notice that despite whatever reasons spawned these wars, the 'cycle of hatred' is always the main factor for these wars. Peace always tends to be temporary, until some country or someone 'snaps' and brings about a new war. However they try to settle things and bring about peace, many people are never satisfied with this, since loved ones and comrades were lost to other nations. This always brings about tensions, and as time goes by and peace 'wears out' the smallest things can spawn new wars.

¹Naruto, chapter 445, page 3