++a is a pre-incrementation. Which means that a is incremented before returning the value of a.
a++ is a post-incrementation. Which means that a is incremented after returning the value of a.
In other words, a++ gives the current value of a and then increment it. While ++a directly increment a. If a=42 then System.out.println(a++) gives 42 while System.out.println(++a) gives 43 and in both cases, a=43 now.
OP also asked for a line by line explanation of that code:
import java.util.Scanner;
public class Number { 
 public static void main(String[] args) { 
   Scanner keyboard = new Scanner(System.in); 
   int number = keyboard.nextInt(); 
   int division1 = (number++) % 10; 
   number = number / 10; 
   System.out.println(number % 10+division1); 
 } 
}
I guess, only the code inside the main function need some explanations :
   // Create a Scanner object that read from the standard input.
   Scanner keyboard = new Scanner(System.in);
   // Read an integer.
   int number = keyboard.nextInt(); 
   // put (number % 10) into variable division1 and then increment `number`.
   int division1 = (number++) % 10;
   // Divide number by 10.
   number = number / 10; 
   // Print that expression :
   System.out.println(number % 10+division1);
The line int division1 = (number++) % 10; might not be very clear. It would be simpler to read like that:
int division1 = number % 10;
number += 1;
Now, the explanation of what the function does:
If number = 142, we put 2 into variable division1, then number is incremented and divided by 10. So number gets the value 14 ((142+1) / 10). And now we print number % 10 + division1 which is 4 + 2 = 6.
Here some examples of results (I've compiled the code myself):
3 => 3
9 => 10
10 => 1
248 => 12