Assuming you actually have a List<AnObject>, all you need is

list.sort(Comparator.comparing(a -> a.attr));

If you make you code clean by not using public fields, but accessor methods, it becomes even cleaner:

list.sort(Comparator.comparing(AnObject::getAttr));

(originally from Ways to sort lists of objects in Java based on multiple fields)

Original working code in this gist

Using Java 8 lambda's (added April 10, 2019)

Java 8 solves this nicely by lambda's (though Guava and Apache Commons might still offer more flexibility):

Collections.sort(reportList, Comparator.comparing(Report::getReportKey)
            .thenComparing(Report::getStudentNumber)
            .thenComparing(Report::getSchool));

Thanks to @gaoagong's answer below.

Note that one advantage here is that the getters are evaluated lazily (eg. getSchool() is only evaluated if relevant).

Messy and convoluted: Sorting by hand

Collections.sort(pizzas, new Comparator<Pizza>() {  
    @Override  
    public int compare(Pizza p1, Pizza p2) {  
        int sizeCmp = p1.size.compareTo(p2.size);  
        if (sizeCmp != 0) {  
            return sizeCmp;  
        }  
        int nrOfToppingsCmp = p1.nrOfToppings.compareTo(p2.nrOfToppings);  
        if (nrOfToppingsCmp != 0) {  
            return nrOfToppingsCmp;  
        }  
        return p1.name.compareTo(p2.name);  
    }  
});  

This requires a lot of typing, maintenance and is error prone. The only advantage is that getters are only invoked when relevant.

The reflective way: Sorting with BeanComparator

ComparatorChain chain = new ComparatorChain(Arrays.asList(
   new BeanComparator("size"), 
   new BeanComparator("nrOfToppings"), 
   new BeanComparator("name")));

Collections.sort(pizzas, chain);  

Obviously this is more concise, but even more error prone as you lose your direct reference to the fields by using Strings instead (no typesafety, auto-refactorings). Now if a field is renamed, the compiler won’t even report a problem. Moreover, because this solution uses reflection, the sorting is much slower.

Getting there: Sorting with Google Guava’s ComparisonChain

Collections.sort(pizzas, new Comparator<Pizza>() {  
    @Override  
    public int compare(Pizza p1, Pizza p2) {  
        return ComparisonChain.start().compare(p1.size, p2.size).compare(p1.nrOfToppings, p2.nrOfToppings).compare(p1.name, p2.name).result();  
        // or in case the fields can be null:  
        /* 
        return ComparisonChain.start() 
           .compare(p1.size, p2.size, Ordering.natural().nullsLast()) 
           .compare(p1.nrOfToppings, p2.nrOfToppings, Ordering.natural().nullsLast()) 
           .compare(p1.name, p2.name, Ordering.natural().nullsLast()) 
           .result(); 
        */  
    }  
});  

This is much better, but requires some boiler plate code for the most common use case: null-values should be valued less by default. For null-fields, you have to provide an extra directive to Guava what to do in that case. This is a flexible mechanism if you want to do something specific, but often you want the default case (ie. 1, a, b, z, null).

And as noted in the comments below, these getters are all evaluated immediately for each comparison.

Sorting with Apache Commons CompareToBuilder

Collections.sort(pizzas, new Comparator<Pizza>() {  
    @Override  
    public int compare(Pizza p1, Pizza p2) {  
        return new CompareToBuilder().append(p1.size, p2.size).append(p1.nrOfToppings, p2.nrOfToppings).append(p1.name, p2.name).toComparison();  
    }  
});  

Like Guava’s ComparisonChain, this library class sorts easily on multiple fields, but also defines default behavior for null values (ie. 1, a, b, z, null). However, you can’t specify anything else either, unless you provide your own Comparator.

Again, as noted in the comments below, these getters are all evaluated immediately for each comparison.

Thus

Ultimately it comes down to flavor and the need for flexibility (Guava’s ComparisonChain) vs. concise code (Apache’s CompareToBuilder).

Bonus method

I found a nice solution that combines multiple comparators in order of priority on CodeReview in a MultiComparator:

class MultiComparator<T> implements Comparator<T> {
    private final List<Comparator<T>> comparators;

    public MultiComparator(List<Comparator<? super T>> comparators) {
        this.comparators = comparators;
    }

    public MultiComparator(Comparator<? super T>... comparators) {
        this(Arrays.asList(comparators));
    }

    public int compare(T o1, T o2) {
        for (Comparator<T> c : comparators) {
            int result = c.compare(o1, o2);
            if (result != 0) {
                return result;
            }
        }
        return 0;
    }

    public static <T> void sort(List<T> list, Comparator<? super T>... comparators) {
        Collections.sort(list, new MultiComparator<T>(comparators));
    }
}

Ofcourse Apache Commons Collections has a util for this already:

ComparatorUtils.chainedComparator(comparatorCollection)

Collections.sort(list, ComparatorUtils.chainedComparator(comparators));

Using Comparator

For Example:

class Score {

    private String name;
    private List<Integer> scores;
    // +accessor methods
}

    Collections.sort(scores, new Comparator<Score>() {

        public int compare(Score o1, Score o2) {
            // compare two instance of `Score` and return `int` as result.
            return o2.getScores().get(0).compareTo(o1.getScores().get(0));
        }
    });

With Java 8 onwards, you can simply use lambda expression to represent Comparator instance.

Collections.sort(scores, (s1, s2) -> { /* compute and return int */ });

Your Comparator would look like this:

public class GraduationCeremonyComparator implements Comparator<GraduationCeremony> {
    public int compare(GraduationCeremony o1, GraduationCeremony o2) {
        int value1 = o1.campus.compareTo(o2.campus);
        if (value1 == 0) {
            int value2 = o1.faculty.compareTo(o2.faculty);
            if (value2 == 0) {
                return o1.building.compareTo(o2.building);
            } else {
                return value2;
            }
        }
        return value1;
    }
}

Basically it continues comparing each successive attribute of your class whenever the compared attributes so far are equal (== 0).

Since Date implements Comparable, it has a compareTo method just like String does.

So your custom Comparator could look like this:

public class CustomComparator implements Comparator<MyObject> {
    @Override
    public int compare(MyObject o1, MyObject o2) {
        return o1.getStartDate().compareTo(o2.getStartDate());
    }
}

The compare() method must return an int, so you couldn't directly return a boolean like you were planning to anyway.

Your sorting code would be just about like you wrote:

Collections.sort(Database.arrayList, new CustomComparator());

A slightly shorter way to write all this, if you don't need to reuse your comparator, is to write it as an inline anonymous class:

Collections.sort(Database.arrayList, new Comparator<MyObject>() {
    @Override
    public int compare(MyObject o1, MyObject o2) {
        return o1.getStartDate().compareTo(o2.getStartDate());
    }
});

Since java-8

You can now write the last example in a shorter form by using a lambda expression for the Comparator:

Collections.sort(Database.arrayList, 
                        (o1, o2) -> o1.getStartDate().compareTo(o2.getStartDate()));

And List has a sort(Comparator) method, so you can shorten this even further:

Database.arrayList.sort((o1, o2) -> o1.getStartDate().compareTo(o2.getStartDate()));

This is such a common idiom that there's a built-in method to generate a Comparator for a class with a Comparable key:

Database.arrayList.sort(Comparator.comparing(MyObject::getStartDate));

All of these are equivalent forms.