The key term to find information pertinent to this question is Zoonosis (link to the Wikipedia article).
However, one should not generalize zoonotic origin to all the diseases affecting humans - in fact, the relationship between humans, diseases, and other organisms carrying these diseases is rather complex.
- Some diseases are uniquely human: their origin is unknown - they might have evolved together with humans or were transmitted to humans or their predecessors millions or billions years ago. Smallpox is one example - it became eradicated as soon as it was eradicated in human population.
- Other diseases have evolved to be transmitted between different types of hosts, e.g., such as humans and mosquitoes. Flaviviruses, such as Zika, Yellow fever, dengue, west Nile virus are examples of such pathogens, adapted to two different types of hosts. In fact, some of these are adapted to mammals other than humans, but capable of infecting humans due to sufficient similarity. Some of these might not be able to effetcively replicate in humans, even though being a serious health hazard.
- For some diseases insects (or other organisms) serve as mechanical vectors, merely carrying disease from one human to another or from a human-like animal to humans, while not infecting the vector.
- There are animals that can be infected by both animal and human viruses. E.g., the types of influenza affecting many birds are not dangerous for humans, and these birds do not get infected with human influenza. However, pigs and paultry may get infected by both. The influenza viruses coexisting in a pig then undergo gene reshuffling, producing new viral strains infectious to humans, that humans get from pigs.
- Finally, what the OP probably had in mind is the transmission of viruses such as Ebola, SARS/MERS, and HIV which exist in nature as animal viruses and suddenly transform to become human viruses. This is an evolutionary process, which is rarely successful.
HIV
E.g., the bushmeat consumption has probably on many occasions introduced SIV to human population - perhaps thousands or millions times during the history of this virus. However, not being adapted to humans, the virus didn't cause pandemics - until the moment when this transmission as accompanied by a fortunate mutation (or mutations) that allowed it successfully replicated and be transmitted in human population. It is quite possible that it took some time for this virus to further adapt to human population before it became as lethal as it is.
Ebola
Ebola is an example of virus that seems to be poorly adapted to human population - although successful in replicating itself, it fails to keep enough hosts alive to assure its permanent existence in the population. Last century has seen multiple Ebola epidemics, all successfully contained by quarantine measures. There have been probably many more spillovers over the course of human history, but they all ended with virus becoming extinct in the human population by eliminating available hosts.