Tl;dr: Joseph Smith's family were important members of early Mormon society, and some held official leadership positions. The church office of Presiding Patriarch was always transmitted by familial inheritance, and this tradition continued until the mid-twentieth century in two of the successor groups that emerged after Smith's death. One of those groups, the Community of Christ, accepted Smith's son as President, and continued to pass that office down by inheritance until 1995. The other group, the LDS Church, has had relatives of Smith serve as Apostle and President, but not as part of an official policy of succession.

Joseph Smith’s family were core supporters of his movement from the very beginning, with his father Joseph, Sr. and his brothers Hyrum and Samuel being among the eleven men who stated that they had seen the gold plates from which the Book of Mormon was transcribed. As the movement grew, several of Joseph's relatives held priesthood positions in the church's governing bodies, the First Presidency and the Quorum of the Twelve Apostles. Most notably, Hyrum was Joseph's Assistant President, and another brother, William, was an apostle. A cousin, George Smith was made an apostle as well, and Joseph, Sr. was Presiding Patriarch (a more ceremonial position), succeeded by Hyrum after his death. Holders of all these offices are considered to be "prophets, seers, and revelators."

For the purposes of this answer I'm focusing on the most prominent leadership positions and positions where familial relationship to Joseph has been required. Other family members were also members of the church, and some brothers and cousins held other positions in the church's organization. Joseph's mother, wife, and sisters were not considered eligible for the priesthood by reason of their gender, but they did occupy a respected position in Mormon society during their lifetimes.

Upon Joseph’s death in 1844, the Assistant President was expected to succeed him, but Hyrum was killed in the same attack as Joseph. Some Mormons expected the next eldest brother, Samuel, to succeed, but he died of an illness one month later. The only remaining brother, William, initially only claimed to have succeeded Hyrum as Presiding Patriarch. Joseph’s eldest son, Joseph III, was considered by some as a possible successor, but he was only 11.

Brigham Young, the President of the Quorum of the Twelve, argued that the Quorum of the Twelve, being considered “equal in authority and power” to the First Presidency, should assume governance. Most of the community supported Young, but some disagreed - William Smith was excommunicated in 1845 after a dispute involving Young's leadership of the church - and there were several competing leadership claims.

In 1847, Young declared that the church members would move from Illinois to Utah. Those who did so became the nucleus of what is now called Church of Jesus Christ of Latter-day Saints, which represents the vast majority of Mormons and is what most people today would consider "the" Mormon church. Members of Smith's family who went to Utah included his cousin George, his uncle John (Joseph Sr.'s brother), Hyrum's wife Mary and their son Joseph F. Smith, and Hyrum's other son "Young" John. Joseph F. Smith went on to become a member of the Quorum of the Twelve and later President of the Church. Descendants of Joseph F. and George Smith have continued to occasionally hold both positions throughout the church's history, but relation to the Smith family has never been a requirement or an automatic qualification to those offices. Additionally, "Uncle" John Smith was made Presiding Patriarch. He was succeeded by "Young" John; Young John's descendants occupied the office until it was discontinued by the church in 1979.

The Smiths who remained in the Midwest included all of Joseph's surviving siblings, his wife Emma, and their children, including Joseph III. In 1860, Joseph III accepted the request of one Mormon group to serve as President. This group became the Reorganized Church of Jesus Christ of Latter-day Saints, today known as the Community of Christ. Over time, the other Smiths in the Midwest became members of this church as well. Joseph III appointed his brother Alexander as Presiding Patriarch. The offices of Presiding Patriarch and President were held by descendants of Joseph Smith until 1958 and 1995, respectively. In both cases, a Smith President nominated the first non-Smith holder of the office.