Jon's answer is great for approaching it from analogy. If a more concrete wording is useful for you, I can pitch in.
Let's start with a variable. A variable is a [named] thing which contains a value. For instance, int x = 3 defines a variable named x, which contains the integer 3. If I then follow it up with an assignment, x=4, x now contains the integer 4. The key thing is that we didn't replace the variable. We don't have a new "variable x whose value is now 4," we merely replaced the value of x with a new value.
Now let's move to objects. Objects are useful because often you need one "thing" to be referenced from many places. For example, if you have a document open in an editor and want to send it to the printer, it'd be nice to only have one document, referenced both by the editor and the printer. That'd save you having to copy it more times than you might want.
However, because you don't want to copy it more than once, we can't just put an object in a variable. Variables hold onto a value, so if two variables held onto an object, they'd have to make two copies, one for each variable. References are the go-between that resolves this. References are small, easily copied values which can be stored in variables.
So, in code, when you type Dog dog = new Dog(), the new operator creates a new Dog Object, and returns a Reference to that object, so that it can be assigned to a variable. The assignment then gives dog the value of a Reference to your newly created Object.
