Output the Partial Products

Pyth, 12 bytes

.e**Qsb^Tk_w

Test suite

Takes input newline separated, e.g.

361
674

Explanation:

.e**Qsb^Tk_w
                Implicit: Q = eval(input()),T = 10
           w    Input the second number as a string.
          _     Reverse it.
.e              Enumerated map, where b is the character and k is the index.
     sb         Convert the character to an int.
   *Q           Multiply by Q.
  *    ^Tk      Multiply by T ^ k. (10 ^ index)

JavaScript (ES7), 48 bytes

(a,b)=>[...b+""].reverse().map((d,i)=>10**i*a*d)

ES6 (56 bytes)

(a,b)=>[...b+""].reverse().map((d,i)=>a*d+"0".repeat(i))

Explanation

Returns an array of partial products as numbers.

(a,b)=>
  [...b+""]    // convert the multiplier to an array of digits
  .reverse()   // reverse the digits of the multiplier so the output is in the right order
  .map((d,i)=> // for each digit d of the multiplier
    10**i      // get the power of ten of the digit
      *a*d     // raise the product of the digit to it
  )

Test

Test uses Math.pow instead of ** to make it work in standard browsers.

var solution = (a,b)=>[...b+""].reverse().map((d,i)=>Math.pow(10,i)*a*d)

A = <input type="text" id="A" value="361" /><br />
B = <input type="text" id="B" value="674" /><br />
<button onclick="result.textContent=solution(+A.value,+B.value)">Go</button>
<pre id="result"></pre>

Jelly, 10 bytes

DU×µLR’⁵*×

Try it online!

How it works

DU×µLR’⁵*×  Left argument: multiplier -- Right argument: multiplicant

D           Convert the multiplier to base 10 (array of digits).
 U          Reverse the array.
  ×         Multiply each digit by the multiplicant.
   µ        Begin a new, monadic chain. Argument: A(array of products)
    L       Get the length of A.
     R      Turn length l into [1, ..., l].
      ’     Decrement to yield [0, ..., l-1].
       ⁵*   Compute 10**i for each i in that range.
         ×  Hook; multiply the powers of ten by the corresponding elements of A.