Perhaps it would help if I tried a more specific example...
Say I have a User. The User is an entity on my system. I'll use a Class to represent that user. The Class will be the blueprint that describes things about the user.
Since each User is different, I can't just use the same Class for every user. I need to make a copy of it that is specific to my User... this is called an Instance.
When you do user = User.new, you're creating a new instance of the User object. This is often referred to as a constructor. (the 'new' method does a bunch of things behind the scenes to prepare the class for later use.)
However, that example is kind of confusing, so let's make it easier to follow. How about
homer = User.new
bart = User.new
both 'homer' and 'bart' are variables that now represent object instances. Assuming that a user had an email address, we would now access that from our variables.
email = homer.email
So to answer your question...
obj = Object.new
obj = variable name and representation of the object instance
= = Assignment operator
Object = class name
. = operator for calling a method in Ruby
new = Class method for Object which initializes a new instance and returns that instance