How many "branches" or different paths of Hinduism are there?

Hinduism is a very disparate religion; it may not even be considered a religion as such, but a meta-religion playing host to many separate religions. 

The term "HInduism" is analogous to "Abrahamism" or even "Jordanism," as the word "Hindu" comes from the Persian pronunciation of Sindhu - the same river that lends its name to the so-called "Indus River Valley civilization." 'Hindu'ism is a convenient umbrella used first by the Persians and later the British, lumping up all of the various astika or quasi astika religions.

Astika means "it is so," and refers to the Vedas; the Veda-affirming religions, as opposed to Carvakas (materialist-atheists), Buddhists and Jains. All of the astika religions claim to represent A. the Vedas, and B. the Sanatana Dharma, which the "true name" for the Hindu religion.

This though is also tricky, as the Sanatana Dharma's self-description is as a perennial religion, always existing and emergent to one degree or another in society. What we have in Hinduism is more the "Hindu Dharma" interpretation of Sanatana Dharma.

Anyway... to come back finally to your question directly, there are hundreds of different sects so standalone as to be just-as-accurately termed separate religions, they are all unified by loose adherence (or claimed adherence) to the Vedas, except for a few minor tantric cults who will actually disparage the Vedas yet are still considered "Hindu" rather than nastika for some odd reason.

They mostly squabble or just stay out of eachother's way, the only actual fighting has been done by nagas - dharma sainiks (warriors) to my knowledge, especially between Vaishnava and Shaiva nagas at the kumbha mela.

The modern Hindu identity is partially a result of a meld which was originally imposed externally and later internalized. In fact, it's one of the reasons now the nation of India exists as such, rather than a loosely associated bloc of small, regional nations centered around particular Indic cultures, languages, and even religions, because after more than a millennia as disparate nations and cultures (with religions being the major unifying force), the political and military might of the Persian invaders forged a pan-Indian culture which the British further shored up after the initial 'divide and conquer,' paving the way for modern self-determination as an expanded nation.