In general an instance variable is local and persisted inside an instance of an object, whereas a local variable is only local and persisted inside a function/object/block scope. For instance:
class User
def name
@name
end
def name= name
@name = name
end
end
def greet user
name = user.name || 'John'
p "Hi, my name is #{name}"
end
user = User.new
greet user
=> 'Hi, my name is John'
name
=> NameError: undefined local variable or method 'name' for main:Object
user.name = "Mike"
greet user
=> 'Hi, my name is Mike'
@name
=> nil
In the greet function name is a local variable that is only defined within that function. The name variable is set on the first line on the function, name = user.name || 'John', but its value is not persisted outside of the function. When you try calling name you get a NameError because name has only been defined as a local variable within the greet function.
@name is local to the user instance of the User class. When you try calling it outside of that context you get nil. This is one difference between local and instance variables, instance variables return nil if they have not been defined, whereas local non-instance variables raise an Error.
Notice that both variable types are local to a specific context though. @name is defined within the user instance, so when you call user.name you are calling the name function in the user instance, in which @name is defined. name is only defined in the greet function, so when you call p "Hi, my name is #{name}" you are able to get a value for name because you are within the scope in which it is defined.