Very briefly, it makes program state unpredictable.
To elaborate, imagine you have a couple of objects that both use the same global variable. Assuming you're not using a source of randomness anywhere within either module, then the output of a particular method can be predicted (and therefore tested) if the state of the system is known before you execute the method.
However, if a method in one of the objects triggers a side effect which changes the value of the shared global state, then you no longer know what the starting state is when you execute a method in the other object. You can now no longer predict what output you'll get when you execute the method, and therefore you can't test it.
On an academic level this might not sound all that serious, but being able to unit test code is a major step in the process of proving its correctness (or at least fitness for purpose).
In the real world, this can have some very serious consequences. Suppose you have one class that populates a global data structure, and a different class that consumes the data in that data structure, changing its state or destroying it in the process. If the processor class executes a method before the populator class is done, the result is that the processor class will probably have incomplete data to process, and the data structure the populator class was working on could be corrupted or destroyed. Program behaviour in these circumstances becomes completely unpredictable, and will probably lead to epic lossage.
EDIT: I also neglected to mention that it Further, global state hurts the readability of your code. If your code has an external dependency that isn't explicitly introduced into the code then whoever gets the job of maintaining your code will have to go looking for it to figure out where it came from.
As for what alternatives exist, well it's impossible to have no global state at all, but in practice it is usually possible to restrict global state to a single object that wraps all the others, and which must never be referenced by relying on the scoping rules of the language you're using. If a particular object needs a particular state, then it should explicitly ask for it by having it passed as an argument to its constructor or by a setter method. This is known as Dependency Injection.
It may seem silly to pass in a piece of state that you can already access due to the scoping rules of whatever language you're using, but the advantages are enormous. Now if someone looks at the code in isolation, it's clear what state it needs and where it's coming from. It also has huge benefits regarding the flexibility of your code module and therefore the opportunities for reusing it in different contexts. If the state is passed in and changes to the state are local to the code block, then you can pass in any state you like (if it's the correct data type) and have your code process it. Code written in this style tends to have the appearance of a collection of loosely associated components that can easily be interchanged. The code of a module shouldn't care where state comes from, just how to process it. If you pass state into a code block then that code block can exist in isolation, that isn't the case if you rely on global state.
There are plenty of other reasons why passing state around is vastly superior to relying on global state. This answer is by no means comprehensive. You could probably write an entire book on why global state is bad.
