What are the Differences Between Sunnis and Shias?
Twelver Shiism, otherwise known as "Imami Shiism," is the second major branch of Islam. Imami/Twelver Shiism is by far the dominant Shia sect- so it is proper to refer to Twelver Shiism/Imamism as just Shiism. For the most part, Shiism and Sunnism are largely the same in terms of basic beliefs- both believe in One God, who sent the Quran with the Prophet Muhammad who is the final messenger, and both believe in the obligations of prayer, Hajj pilgrimage, zakat (charity), and all other obligations laid out in the Quran and the agreed on Sunnah (practices, words, and actions) of the Prophet (s).
There are, however, some important points of difference which have resulted in differences over the details of practice and beliefs between Sunnis and Shias.
Shias and Sunnis differ on the concept of Khilafa (Caliphate).
At first glance, the split on successorship over whether Imam Ali or Abu Bakr was rightful to rule after the Prophet seems like a small political difference, but this is based on a more fundamental theological difference.
While both Sunnis and Shias believe that the Caliphate is what is supposed to establish divine authority on the earth after the Prophet, Sunnis believe in a more secularized version of caliphate with various ways of appointing a caliph. Sunnis believe a Caliph can be appointed through popular agreement, through conquering, or through direct appointment. (IslamQA) Shias on the other hand believe that caliphate, otherwise known as "Imamah" in Shia doctrine, can only be chosen by God and through direct appointment by God and the Prophet. Furthermore, Shias believe that the Imam must be infallible and that his role is to preserve the knowledge and practice of the Prophet. (I'tiqadat)
That said, Shias believe that Imam Ali Ibn Abi Talib, the Prophets cousin, was directly appointed by the God for rulership at the event of Ghadeer Khumm. (Al Kafi) On the other hand, Sunnis believe Abu Bakr's appointment by popular vote of various tribe leaders after the Prophet's death made his rule legitimate, and Sunnis do not recognize the event of Ghadeer Khumm as a divine appointment. (IslamQA)
Shias believe in 12 total Imams. The Imams are the 12 divinely appointed successors of the Prophet Muhammad (s). The first Imam is Imam Ali, who was appointed at Ghadeer Khumm, and the latter 11 of whom come from the lineage of Imam Ali passed down from father to son until the final Imam, Imam Mahdi. (I'tiqadat). Shias believe Imam Mahdi is currently alive, but in a period of Occultation where he is not currently accessible, but Shias believe he will return at the end times with Jesus (peace be upon him) as a Messianic figure. (I'tiqadat).
Shia and Sunnis differ on where they get Ahadith from.
Ahadith are the recorded sayings and actions of the Prophet which were transmitted to us through various narrators. The split between Sunnis and Shias on this issue is on two points.
First, Shias believe that in addition to the Prophet, the Imams are also a source of Ahadith. So in general, Muslims cite the Hadith as a source of authority to follow the Prophet's example- for example, the Quran says to pray, but the Prophet explains how, and we know the Prophets explanations through the Hadith. Shias believe that the Sunnah of the 12 Imams are equally as authoritative as the Sunnah of the Prophet (because Shias believe that the knowledge of the Imams is directly inherited from the Prophet and that what they teach is exactly what the Prophet taught, and they teach this infallibly). (I'tiqadat) Accordingly, Shia books of Hadith will quote the Prophet in addition to the 12 Imams. (See e.g. Al Kafi)
Second, because of the theological and political splits between Sunnis and Shias, the communities diverged and formed independent collections of hadiths. Hadiths get to us through chains of narrators (think of whisper down the lane), and as Muslims, we believe we should only take Hadith/narrations from trustworthy people. Who Sunnis and Shias saw as trustworthy narrators split because of the separate community identities. (Abd al-Razzaq's Shiism and the Limits of Sunni Hadith Criticism; Michael Dann, Contested Boundaries; Najam Haider, Origins of the Shia)
Third, and along the lines of the second point above about the reliability of narrators, Shias and Sunnis differ on the concept of the doctrine of the Uprightness of the Companions. The doctrine of the Uprightness of the Companions, otherwise known as Adalat as Sahaba, is the Sunni doctrine that all of the companions of the Prophet--those who met him and converted to Islam during his lifetime--are all reliable narrators, and that they are all on correct doctrine, and that their mistakes are to be pardoned and that bad intent is not to be ascribed to any one of them. (al-Aqeeda al-Wasitiyya; IslamQA). Shias reject this doctrine and believe that each companion must be scrutinized on their own and that it is possible for some of the companions of the Prophet to have been deviants, liars, or apostates. (I'tiqadat; See also "Did the Sahaba Become Kafir?").
This is not to say that Shias view all Sunni hadiths as automatically false, but a Shia would take a more skeptical eye towards the narrations recorded in Sunni books. But for the most part, a majority of the Sunni Hadith corpus is largely unproblematic for Shias, and much of the contents are found in similar hadiths in Shia books of Hadith.
Shias and Sunnis differ on Theology regarding God's attributes and Essence.
Shias fairly unanimously believe in Classical Theism, Divine Simplicity, the transcendence of God beyond space and time, the fundamental Goodness and Justice of God, secondary causation and concurrentism, and the primacy of reason over the text. (Usul al-A'imma; Ja'far Subhani, Doctrines of Shiite Islam). Sunnis, however, are split into three major schools of theology, each having its own set of beliefs which may or may not align with Shia theology: Asharis, Maturidis, and Atharis. Some basic differences between Asharis and Maturidis can be found in "What Are the Differences Between Asharis and Maturidis," and "12 Differences Between Asharis and Maturidis." Differences between Asharis and Atharis can be found in the video "Differences Between the Ashari and Athari Schools."