Static vs non-static class members
You need to think about static variables as belonging to the class, not to instances of the class.
If, in all instances of the class this variable should be identical, use a static variable.
If not, use an instance variable.
In general having public static variables is bad practice - it is a shared global resource and if you change it you need to synchronize access to it. Having global state is something you want to avoid as much as possible.
Best practice is to avoid public static. In OOP, class is meant to hide its members. Static is actually not a member of the instance but of the type.
Static comes handy if you are implementing singleton pattern. But then again they need to be made private and accessible through a public property.
You need to read Static Classes and Static Class Members (C# Programming Guide).
Well I can't conclusively say that one is better, because they serve different purposes.
Are you familiar with OOP? In OOP, static objects or members of a class that can be accessed directly from the class, while non-static members can only be accessed from the instance it belongs to.
C# follows a similar principle for the methods. The static methods can by accessed directly from the class, while non-static methods (or instance methods as I like to call them) have to be accessed from an instance. That is why instatiating needs to be done for instance methods, while for static methods it's just not needed, and furthermore impractical (see below).
In OOP, static variables are used for values which cannot be stored by an instance variable. Example: supposed you wanted to keep a count of how many instances of a class exists? How would you store that in a single instance?
The methods use a similar principle. They should be used for procedures for which it is impractical to do within an instance of a class. I tend to use them for broad procedures (not a technical term), meaning those that do not require me to instantiate an object. Example, adding two parameters. (This usage may or may not be correct, but I believe it is)
However, if you wanted to add two properties of an object, the method cannot be static, because as you would soon realize, static methods cannot access instance methods or variables within a class. Of course that makes sense because that static method would not know which instance of the class the get these from unless it were told, since it is not part of an instance itself)
For the sake of no further complicating things, I'll stop here. Let me know if you misunderstood anything.
Your choice depends on your architecture.
Static makes part of a Type, others make part of an instance of that type. If you want have some shared state (say) between different instances of the same type, use static. If you want that every instance have it's own value, independent from others, use instance fields.
In both cases, by the way, avoid to expose like a public fields, but use properties.
I completely agree with Mr Oded:
If, in all instances of the class this variable should be identical, use a static variable.
If not, use an instance variable.
Yes, adding static to a class member basically means you can access it without an instance, and only outside any instance. And yes, it becomes a global resource, or even a global variable if you will.
But I think there's at least another (heavily edited) good point to be made here...
Using static members as global vars go against OOP
This means once you set a static member you can't pass it around as an object. The more you use static as global var, the more difficult it is for unit testing / mocking classes.
There is a solution for that, Singletons. But they should never come without warnings!
At other hand, if you're sure you really need global vars, take a look at the Toolbox pattern. It's a not well known extension of Singleton pattern. It's so unknown in fact, if you google for it you won't find it with those keywords (toolbox pattern).
So plan ahead. Read more. Get to know about every option so you can decide better. Even get a book. Object Oriented Programming is more about applying concepts that will help in the long run than just making things work now.