With Java 8:

Comparator.comparing((Person p)->p.firstName)
          .thenComparing(p->p.lastName)
          .thenComparingInt(p->p.age);

If you have accessor methods:

Comparator.comparing(Person::getFirstName)
          .thenComparing(Person::getLastName)
          .thenComparingInt(Person::getAge);

If a class implements Comparable then such comparator may be used in compareTo method:

@Override
public int compareTo(Person o){
    return Comparator.comparing(Person::getFirstName)
              .thenComparing(Person::getLastName)
              .thenComparingInt(Person::getAge)
              .compare(this, o);
}

First, all the examples you say cause errors compile fine with the reference implementation (javac from JDK 8.) They also work fine in IntelliJ, so its quite possible the errors you're seeing are Eclipse-specific.

Your underlying question seems to be: "why does it stop working when I start chaining." The reason is, while lambda expressions and generic method invocations are poly expressions (their type is context-sensitive) when they appear as method parameters, when they appear instead as method receiver expressions, they are not.

When you say

Collections.sort(playlist1, comparing(p1 -> p1.getTitle()));

there is enough type information to solve for both the type argument of comparing() and the argument type p1. The comparing() call gets its target type from the signature of Collections.sort, so it is known comparing() must return a Comparator<Song>, and therefore p1 must be Song.

But when you start chaining:

Collections.sort(playlist1,
                 comparing(p1 -> p1.getTitle())
                     .thenComparing(p1 -> p1.getDuration())
                     .thenComparing(p1 -> p1.getArtist()));

now we've got a problem. We know that the compound expression comparing(...).thenComparing(...) has a target type of Comparator<Song>, but because the receiver expression for the chain, comparing(p -> p.getTitle()), is a generic method call, and we can't infer its type parameters from its other arguments, we're kind of out of luck. Since we don't know the type of this expression, we don't know that it has a thenComparing method, etc.

There are several ways to fix this, all of which involve injecting more type information so that the initial object in the chain can be properly typed. Here they are, in rough order of decreasing desirability and increasing intrusiveness:

• Use an exact method reference (one with no overloads), like Song::getTitle. This then gives enough type information to infer the type variables for the comparing() call, and therefore give it a type, and therefore continue down the chain.
• Use an explicit lambda (as you did in your example).
• Provide a type witness for the comparing() call: Comparator.<Song, String>comparing(...).
• Provide an explicit target type with a cast, by casting the receiver expression to Comparator<Song>.

is Comparator.comparing() used for converting a single argument lambda expression to a double argument?

Yes, you can sort of think of it like that.

When sorting things, you are supposed to specify "given two things a and b, which of them is greater, or are they equal?" using a Comparator<T>. The a and b is why it has 2 lambda parameters, and you return an integer indicating your answer to that question.

However, a much more convenient way to do this is to specify "given a thing x, what part of x do you want to sort by?". And that is what you can do with the keyExtractor argument of Comparator.comparing.

Compare:

/*
given two people, a and b, the comparison result between a and b is the 
comparison result between a's name and b's name
*/
Comparator<Person> personNameComparator = 
    (a, b) -> a.getName().compareTo(b.getName());

/*
given a person x, compare their name
*/
Comparator<Person> personNameComparator = 
    Comparator.comparing(x -> x.getName()); // or Person::getName

The latter is clearly much more concise and intuitive. We tend to think about what things to sort by, rather than how exactly to compare two things, and the exact number to return depending on the comparison result.

As for the declaration for comparing:

public static <T, U extends Comparable<? super U>> Comparator<T> comparing(
            Function<? super T, ? extends U> keyExtractor)

The <T, U extends Comparable<? super U>> part first declares two generic type parameters - T is what the comparator compares (Person in the above case), and U is the type that you are actually comparing (String in the above case), hence it extends Comparable.

keyExtractor is the parameter you pass in, such as x -> x.getName(), that should answer the question of "when given a T, what is a U that you want to compare by?".

If you are confused by the ? super and ? extends, read What is PECS?.

If you haven't realised already, the implementation of comparing basically boils down to:

return (a, b) -> keyExtractor.apply(a).compareTo(keyExtractor.apply(b));