How does the execution of a particular test method depend on other test
methods in TestNG?

The execution of a particular test method can be made dependent on another test method with the help of dependsOnMethods helper attribute.

Example

@Test(dependsOnMethods={"Payment"})
public void verifyLoan(){
   System.out.println("Loan payment successful");
}
@Test
public void Payment(){
   System.out.println("Payment successful ");
}
@Test
public verifyTransaction(){
   System.out.println ("Transaction verification");
}

Here in the Java class file, verifyLoan() method will only be executed after the Payment() method is run successfully. But verifyTransaction() method runs independently without having a precondition test method to be executed.