String replace method is not replacing characters

Strings are immutable, meaning their contents cannot change. When you call replace(this,that) you end up with a totally new String. If you want to keep this new copy, you need to assign it to a variable. You can overwrite the old reference (a la sentence = sentence.replace(this,that) or a new reference as seen below:

public class Test{

    public static void main(String[] args) {

        String sentence = "Define, Measure, Analyze, Design and Verify";

        String replaced = sentence.replace("and", "");
        System.out.println(replaced);

    }
}

As an aside, note that I've removed the contains() check, as it is an unnecessary call here. If it didn't contain it, the replace will just fail to make any replacements. You'd only want that contains method if what you're replacing was different than the actual condition you're checking.

And when I debug this the logic does fall into the sentence.replace.

Yes, and then you discard the return value.

Strings in Java are immutable - when you call replace, it doesn't change the contents of the existing string - it returns a new string with the modifications. So you want:

sentence = sentence.replace("and", " ");

This applies to all the methods in String (substring, toLowerCase etc). None of them change the contents of the string.

Note that you don't really need to do this in a condition - after all, if the sentence doesn't contain "and", it does no harm to perform the replacement:

String sentence = "Define, Measure, Analyze, Design and Verify";
sentence = sentence.replace("and", " ");

You aren't doing anything with the return value of replace. You'll need to assign the result of the method, which is the new String:

sentence = sentence.replace("and", " ");

A String is immutable in java. Methods like replace return a new String.

Your contains test is unnecessary: replace will just no-op if there aren't instances of the text to replace.

You should re-assign the result of the replacement, like this:

 sentence = sentence.replace("and", " ");

Be aware that the String class is immutable, meaning that all of its methods return a new string and never modify the original string in-place, so the result of invoking a method in an instance of String must be assigned to a variable or used immediately for the change to take effect.