Why is this C program showing any random integer other than expected?

To add to what Greg has - rightly - said, compiling does not mean your code is right! :laugh:
Think of the development process as writing an email: compiling successfully means that you wrote the email in the right language - English, rather than German for example - not that the email contained the message you wanted to send.

So now you enter the second stage of development (in reality it's the fourth or fifth, but you'll come to the earlier stages later): Testing and Debugging.

Start by looking at what it does do, and how that differs from what you wanted. This is important, because it give you information as to why it's doing it. For example, if a program is intended to let the user enter a number and it doubles it and prints the answer, then if the input / output was like this:

Input   Expected output    Actual output
  1            2                 1
  2            4                 4
  3            6                 9
  4            8                16

Then it's fairly obvious that the problem is with the bit which doubles it - it's not adding itself to itself, or multiplying it by 2, it's multiplying it by itself and returning the square of the input.
So with that, you can look at the code and it's obvious that it's somewhere here:

int Double(int value)
   {
   return value * value;
   }

Once you have an idea what might be going wrong, start using the debugger to find out why. Put a breakpoint on the first line of the method, and run your app. When it reaches the breakpoint, the debugger will stop, and hand control over to you. You can now run your code line-by-line (called "single stepping") and look at (or even change) variable contents as necessary (heck, you can even change the code and try again if you need to).
Think about what each line in the code should do before you execute it, and compare that to what it actually did when you use the "Step over" button to execute each line in turn. Did it do what you expect? If so, move on to the next line.
If not, why not? How does it differ?
Hopefully, that should help you locate which part of that code has a problem, and what the problem is.
This is a skill, and it's one which is well worth developing as it helps you in the real world as well as in development. And like all skills, it only improves by use!

Thta's not the proper way to compute the maximum. Try

#include <stdio.h>
  
enum
{
  MAX_ITEMS = 100,
};

int main(void)
{
  int integers[MAX_ITEMS];
  int items;

  for (items = 0; items < MAX_ITEMS; ++items)
  {
    printf("Enter the integer (0 to terminate): \n");
    scanf("%d", &integers[items]);

    if (integers[items] == 0)
      break;
  }

  if ( items > 0 )
  {
    int max = integers[0];
    for(int i = 1; i < items; ++i)
    {
      if ( max < integers[i] )
        max = integers[i];
    }
    printf("The max is %d\n", max);
  }
  return 0;
}

if (items > 0)
  {
  ...    
  }
else
  puts("no input");

Quote:

Correct me for where I am wrong. This is giving some random integer instead of entered ones.

In C++, you are responsible of everything.

int integers[100];

zeroing the array is your responsibility !
If your input is 5 integers, what is the value of the 95 others in the array?
Because no matter what is the input, you search max on the whole array.

What about finding the problem by yourself ?
Your code do not behave the way you expect, or you don't understand why !

There is an almost universal solution: Run your code on debugger step by step, inspect variables.
The debugger is here to show you what your code is doing and your task is to compare with what it should do.
There is no magic in the debugger, it don't know what your code is supposed to do, it don't find bugs, it just help you to by showing you what is going on. When the code don't do what is expected, you are close to a bug.
To see what your code is doing: Just set a breakpoint and see your code performing, the debugger allow you to execute lines 1 by 1 and to inspect variables as it execute.

Debugger - Wikipedia, the free encyclopedia[^]

Mastering Debugging in Visual Studio 2010 - A Beginner's Guide[^]
Basic Debugging with Visual Studio 2010 - YouTube[^]

1.11 — Debugging your program (stepping and breakpoints) | Learn C++[^]

The debugger is here to only show you what your code is doing and your task is to compare with what it should do.

Quote:

Correct me for where I am wrong.

The hints from Greg, CPallini and Patrice are already mentioned.

I write down everything again and add:

-The field integers is not initialized, there are random values in it.
-You abort the reading, but then evaluate (more than!) 100 values.
-It can be that there are no values at all, it is nevertheless continued.
-In the second for-loop the value [a+1] must not be used, because a=99.

The compiler still points out:
-to use the scanf_s() function instead of scanf().
-warning C4477: "printf": The format string "%d" requires an argument of type "int