Typically, you'd write a unit test program that creates some objects, sets them up, and use asserts to verify conditions that you are expecting to be true. The program will alert you when an assertion fails.
So in your test program you could, for example:
CheckingAccount test = new CheckingAccount(1);
CheckingAccount other = new CheckingAccount(2);
SavingAccount anotherTest = new SavingAccount();
SavingAccount anotherOther = new SavingAccount();
anotherTest.accountNumber = 3;
anotherOther.accountNumber = 3;
assert !test.equals(other); // this should evaluate to true, passing the assertion
assert anotherTest.equals(anotherOther); // this should evaluate to true, passing the assertion
It looks like you use an account number as a means of equality for your accounts, so I'm assuming when creating these objects, you either pass the account number as a parameter for the constructor, or assign it explicitly
Obviously this is a very meager example, but I'm not sure about the creation/structure of your objects. But this could be extended to provide more meaningful testing, as long as you get the gist.
EDIT so to fully test your equals method, you can set up your assertions so that they all should evaluate to true (and pass) as well as testing all the functionality of your equals method (complete code coverage)
CheckingAccount newTest = new CheckingAccount(1);
CheckingAccount secondTest = new CheckingAccount(1);
SavingAccount newOther = new SavingAccount(3);
assert newTest.equals(newTest); // test first if
assert !newTest.equals(null); // test second if
assert !newTest.equals(newOther) // test third if
assert newTest.equals(secondTest); // test fourth if