The Senju clan was descended from the younger of the two brothers chosen by the Sage of the Six Paths to inherit the way of the ninja. He was born with the "body" of the sage, and he believed that love was the key to peace. This led to the sage choosing him as his successor causing his older brother to attack him out of hatred. Eventually, the Uchiha clan was descended from this older brother and resulted in the bitter rivalry of the Senju and Uchiha.
The Senju clan gained their fame in the era before the founding of the hidden villages. Where other clans focused on mastering one particular skill, the Senju were masters in all skills, from ninjutsu, to taijutsu, to genjutsu. Because of this, they were feared as "the clan with a thousand skills" (「千の手を持つ一族」, "sen no te o motsu ichizoku"), which gave them their name Senju (千手, "a thousand skills", "a thousand hands").
The most famous member of the clan was their leader, Hashirama Senju. His unique Wood Release ninjutsu even gave the clan the nickname the Senju clan of the Forest (森の千手一族, Mori no Senju Ichizoku). This Wood Release, and his ability to control the tailed beasts, made him the greatest shinobi of his time.
The Senjus' place at the top of the shinobi world was undisputed, but there was one clan that was able to stand up to them: the Uchiha clan. Whenever one country hired the Senju, their enemies would hire the Uchiha, and vice versa. This led to a rivalry between the two clans and, more specifically, between the leaders of the clans, Hashirama and Madara Uchiha.
However, in a world of constant bloodshed, Hashirama had a dream of peace, and a great love for his fellow men. Known as the "Will of Fire" (火の意志, "Hi no Ishi"), this belief made the Senju clan propose a truce to the Uchiha clan. Weary of fighting, and persuaded by Hashirama's great charisma and negotiation abilities, the Uchiha accepted, ending their bloody rivalry. Only Madara was against this truce, believing the Senju would one day destroy the Uchiha clan. He was forced to accept, but his humiliation wouldn't end there.
Some time after the alliance between the Senju and the Uchiha was signed, the Land of Fire hired them to stabilize its borders, and the clans created a permanent settlement within the country. Thus, Konohagakure was founded, and, with it, the system of one Hidden Village per country. As the system spread throughout the world, the constant violence stopped, and Hashirama's dream began to become a reality. But it would soon start anew after Hashirama was elected as the First Hokage; a move Madara felt was a humiliation, and made him decide to leave the new village, eventually returning to attack it.
Throughout the years, the Senju retained their supremacy over Konoha, and made the village into what it is today. After the First Hokage's death, his younger brother, Tobirama Senju, stepped up to become the Second Hokage. Although likely not a Senju by blood, Hiruzen Sarutobi, who would become the Third Hokage after the Second's death, was trained by both the First and Second. His successor, Minato Namikaze, was the student of Jiraiya, who was a student of the Third. Tsunade, the Fifth Hokage, was not only a student of the Third, but the granddaughter of the First and the grandniece of the Second. All of them strongly held to the Will of Fire philosophy.
It is unknown if the Senju clan currently still exists as an organization. There are no known people with the Senju name currently alive, although the Fifth Hokage, Tsunade, might be an official Senju. It is very possible that the Senju clan disbanded into smaller clans and families, especially since they never had a specific skill or technique like the Uchiha or Hyūga clans. However, their legacy lives on strongly in the hearts of the people of Konoha.