In ruby, operators are in fact method calls. If you have two variables a and b and want to check their equality, you generally write a == b, but you could write a.==(b). The last syntax shows what happens during an equality check : ruby calls a's method == and passes it b as an argument.
You can implement custom equality check in your classes by defining the == and/or the eql? methods. In your example, other is simply the name of the argument it receives.
class Person
attr_accessor :name
def initialize name
@name = name
end
end
a = Person.new("John")
b = Person.new("John")
a == b # --> false
class Person
def == other
name == other.name
end
end
a == b # --> true
For your second question, the only methods starting with == you're allowed to implement are == and ===. Check here for the full list of restrictions on method names in ruby: What are the restrictions for method names in Ruby?