Look, up in the sky! It's a super duper array!

Jelly, 47 45 4̷4̷ 42 bytes

+\,×\<ḊZṖP“sd”x;€“uper ”;/“It's a ”,“¥ṫɲ»j

This prints an empty string (falsy) for arrays that are neither super nor duper. Try it online!

How it works

+\,×\<ḊZṖP“sd”x;€“uper ”  Main link (first half). Argument: A (array)

+\                        Compute all partial sums of A.

   ×\                     Compute all partial products of A.
  ,                       Pair the results to the left and to the right.
     <Ḋ                   Perform vectorized comparison with A[1:].
                          This yields a 2D array of Booleans.
       Z                  Zip; pair the Booleans corresponding to each integer.
        Ṗ                 Remove the last pair.
                          (Nothing is compared with the last sum/product.)
         P                Take the product of each column.
          “sd”x           Perform vectorized character repetition.
                          This yields ['s', d'], ['s'], ['d'], or [].
               ;€“uper ”  Append the string "uper " to each character.


;/“It's a ”,“¥ṫɲ»j        Main link (second half).

;/                        Reduce the resulting array of strings by concatenation.
                          This will fail for an empty array, exiting immediately.
  “It's a ”,“¥ṫɲ»         Push ['It's a ', 'array!'].
                 j        Join that array, separating by the super duper string.

JavaScript (ES6), 111 110 bytes

Saved a byte thanks to @ETHproductions!

a=>a.map((n,i)=>i&&(s=s&&n>s&&s+n,d*=d&&n>d&&n),s=d=a[0])|s|d&&`It's a ${s?"super ":""}${d?"duper ":""}array!`

Explanation

Takes an array of numbers, returns a string or the number 0 for false.

a=>
  a.map((n,i)=>      // for each number n at index i
    i&&(             // skip the first number (because s and d are initialised to it)
      s=s&&n>s&&s+n, // if it is still super and n > s, s += n, else s = false
      d*=d&&n>d&&n   // if it is still duper and n > d, d *= n, else d = false
    ),
    s=               // s = sum of previous numbers if super, else false
    d=               // d = product of previous numbers if duper, else false
      a[0]           // initialise s and d to the first number
  )
  |s|d               // if it is neither super or duper, output 0

  // Output the appropriate string
  &&`It's a ${s?"super ":""}${d?"duper ":""}array!`

Test

var solution = a=>a.map((n,i)=>i&&(s=s&&n>s&&s+n,d*=d&&n>d&&n),s=d=a[0])|s|d&&`It's a ${s?"super ":""}${d?"duper ":""}array!`

Numbers (space-separated) = <input type="text" id="input" value="2 3 7 43 1856" />
<button onclick="result.textContent=solution(input.value.split(' ').map(n=>+n))">Go</button>
<pre id="result"></pre>

Java, 183 182 Bytes

String w(int[]a){boolean s=1<2,d=s;int m=a[0],p=m,k=a.length,i=0,e;if(k>0)for(;++i<k;s&=e>m,d&=e>p,m+=e,p*=e)e=a[i];return d|s?"It's a "+(s?"super ":"")+(d?"duper ":"")+"array!":"";}

I made the following assumptions:

• The output is via return value.
• The empty String "" is a falsy value.

If any of these are wrong, please tell me.

Anyway, I can't shake the feeling that I might have gone overboard with the amount of variables.

Edit: managed to save a byte, thanks to @UndefinedFunction