How to continue execution when Assertion is failed

I suggest you to use soft assertions, which are provided in TestNg natively

package automation.tests;

import org.testng.asserts.Assertion;
import org.testng.asserts.SoftAssert;

public class MyTest {
  private Assertion hardAssert = new Assertion();
  private SoftAssert softAssert = new SoftAssert();
}

@Test
public void testForSoftAssertionFailure() {
  softAssert.assertTrue(false);
  softAssert.assertEquals(1, 2);
  softAssert.assertAll();
}

Source: http://rameshbaskar.wordpress.com/2013/09/11/soft-assertions-using-testng/

Selenium IDE uses verify to perform a soft assertion, meaning that the test will continue even if the check fails and either report the failures at the end of the test or on the event of a hard assertion.

With TestNG it is possible to have these soft assertions by using custom test listeners. I have documented how to do this on my blog: http://davehunt.co.uk/2009/10/08/using-soft-assertions-in-testng.html

Basically, you need to create your own verify* methods, in these you can catch assertion failures and add them to a map. Then in a custom afterInvocation listener you can set the test to failed if the map is not empty.

I am adding again one of the easiest ways to continue on assertion failure. This was asked here.

try{
        Assert.assertEquals(true, false);
        }catch(AssertionError e)
        {
            System.out.println("Assertion error. ");
        }

        System.out.println("Test Completed.");

Change your assertions to verifications:

verifyTrue(selenium.isTextPresent("Please enter Email ID"));
verifyTrue(selenium.isTextPresent("Please enter Password"));