"Hinduism", like "Buddhism", is a very loose term. There are many many many schools of Hinduism and none of these schools have been static. This is a huge question (I know, most interesting ones are) but let me try to at least outline some of the branches of one of the schools, the Vedanta school, one of the most prominent schools. The name Vedanta refers to the end of the Veda scriptures: the Upanishads, the Brahma Sutras, and the Bhagavad Ghita.

Nondualism

Shankara, the prominent figure in this thought, lived in the 8th and 9th centuries. Born to the Brahmin caste (not to be confused with Brahman), Shankara lead an ascetic life starting at a very early age and is also responsible for founding four famous "monasteries" (mathas) in India, one in each cardinal direction (north, south, east, west). Most of his writings were commentaries on the texts that the vedanta schools focus on, the Upanishads, the Bhagavad Gita, and the Brahma Sutras (one of many Sutras, such as the Lotus Sutra, or the Kama Sutra). There are three points that Shankara outlines all his thinking with, underlining his belief in non dualism (that there is only one reality and that everything is a part of it): Only Brahman is real, the world is an illusion, and the self is part of Brahman. So, all change in this world is an illusion/distortion, and there is no creation, everything is just part of Brahman. There's this kind of partial reality vs ultimate reality position that is reflected in many points. For example, as for the ultimate reality of Brahman argued by Shankara, it is "God" without quality or attributes, just pure existence and consciousness and beyond any names or forms. Additionally, Atman (the "soul") is all part of Brahman, and it is only the distorted reality that keeps them separate. To break the cycle of reincarnation, samsara, Shankara puts forth that a kind of "self correcting" and logical thinking and wisdom are aids towards enlightenment, but only partial (the analogy is "this way is like a knife that can cut everything around it, but cannot cut itself"). It is only through "non thinking" that this can be achieved, with the help of aided study. A great and very famous analogy is that it is like 10 young men who make a river crossing. After crossing, one of them counts the group, and only counts 9. Another does the same, and also only counts 9. A passerby sees what's going on, and notices that each had counted all but himself, and points to each, saying "you are the tenth one". And thus, they realized that they were that, and always were.

Qualified Dualism

Ramanuja comes around in the 11th or 12th century (there's some debate). He also became a Vaishnavite monk, and wrote commentaries on the same works as Shankara. But the conclusions drawn were not the same. Ramanuja felt that the Brahman defined by Shankara was too remote and abstract. Basically, it was very hard to incorporate temple Bhakti to Shankara's idea of Brahman. Ramanuja places vast importance on bhakti with an individual god, and also argues for qualified nondualisim. In Ramanuja's argument, Atman and Brahman are the same substance, but are of different qualities. I hesitate to make this comparison, but it's a bit like the Nicene Creed, in that it resolves apparent contradictions. So there's one reality of Brahman, but there are distinct parts and does change, just within Brahman. Additionally, the world is not an illusion, it is a reality, and souls exist and are distinct, but are not entirely independent. You can distinguish between the "you" and "that", but they're both ultimately the same. It might be a little confusing, so don't worry. In Ramanuja's idea of samsara, again, the escaping of the reincarnation cycle, instead of "returning" as part of Brahman, your soul escapes, and basks within the presence of Brahman. Another difference between Shankara and Ramanuja, is that while Shankara says the main problem is ignorance, and the solution is learning, Ramanuja argues that the main problem is unbelief, and the solution is bhakti. This emphasis on bhakti as being the way of salvation/enlightenment spawns a great amount of poetry as a form of devotion/bhakti and a lot of it is written as love poetry (think the Song of Solomon). Finally, caste, merit, and even religion don't impact one's reunion with Brahman.

Dualism

Madhva comes in the late 12th, early 13th century and was part of the Brahmin caste, same as Shankara, and wrote commentaries on the same vedanta texts. Madhva is not a non dualist. Instead, Madhva is solidly a dualist and is the founder of dualistic vedanta. Madhva argued that there were two things, separate: Brahman and Atman. Additionally, he put forth that Brahman, matter, and self all are distinct and permanent, but matter and self are subordinate and dependent on the self-dependant Brahman. In Madhva's school, the focus is on the differences between Brahman and self, rather than the unity between the two. Brahman is different than self, Brahman is different than matter, matter is different than self, and self is different from self. Self takes on several different qualities, some free of samsara, some passing through samsara, and others will eternally continue through samsara (this last one is a completely new idea). To break out of samsara, one participates in scripture study and gains knowledge, and then Brahman lifts the veil, as a sort of salvation half by works, half by bestowment. Madhva's thinking also opens up much room for theism, as the Brahman described by Madhva has very clear characteristics, as opposed to, say, Shankara's idea of Brahman.