Reading this wikipedia article is strongly recommended. It is precisely about the topic you are researching about.
A river can split due to natural/geologic activities or people can split it for their own reasons.
1- A very wide river can split due to volcanic action. Orogenic events can push a series of hills through a riverbed, splitting it into two (or more) branches.
2- Rivers can also be split due to massive landslides and mudflows. However, these types of divisions are temporary, as the constant flow of water dissolves or displaces the mud and other obstacles in its way.
3- With modern technology, it is possible for people to change the course of, or split a river. Filling the middle part of a river with massive rocks and then stabilizing them with metallic supports would split a river. One of the branches can then be used to provide water to a city or utilized for irrigation.
There are many examples of river branches which are permanently coexisting. These examples are listed in the article linked above. Tributaries usually dry up due to geologic events, when there is some major geological activity at the place of the split, which stops or discourages the flow of water into the channel. Most of the time, this happens when the mouth of the tributary is elevated due to orogenic events or when excessive amounts of rocks break from a nearby mountain and shut the mouth of the tributary.