#Java, 123 bytes

    interface y{static void main(String[]s){int a=new Integer(s[0]),c=0,i;while(++c<=a){i=c;while(i>0)System.out.print(i--);}}}

Takes input as command line argument.

Ungolfed with comments:

    interface y{
        static void main(String[] s) {
            int a = new Integer(s[0]),          // Input number
                    c = 0,                      // Counter 1
                    i;                          // Counter 2
            while (++c <= a) {                  // Increment counter 1. Until counter 1 is less or equal to input
                i = c;                          // Counter 2 = counter 1
                while (i > 0)                   // Until counter 2 is more than 0
                    System.out.print(i--);      // Output counter 2 and decrement it
            }
        }
    }