To answer your comment, yes there are equivalent of PEP8 in java. I suggest checkstyle, this plugin works in lots of IDE / text editors and can be configured with your code style.

Code review

reverse_string method

1. In java, we try to use the upper-case version of Snake case only on constants, for / enums. For the methods and variablevariables, I suggest that you use the Camel case style.

Before

private static String reverse_string(String my_string) 
{
   //[..]
}

After

private static String reverseString(String myString) 
{
   //[..]
}

2. When concatenating String in a loop, it's generally recommended to use a java.lang.StringBuilder to gain performance and take fewer operations.

private static String reverseString(String myString) 
{
   StringBuilder reversedString = new StringBuilder();

   for (int j = myString.length() - 1; j >= 0; j--) 
   {
      reversedString.append(myString.charAt(j));
   }

   return reversedString.toString();
}

If you want not to use it, then I suggest the += operator instead of the +; it will give the same result and make the code shorter.

Other observations

The code was missing a bit of formatting, but nothing too important, I sugest that you pick a formatter for your style (Horstmann style) depending on your IDE / text editor.

Refactored code

private static String reverseString(String myString) 
{
   String reversed_stringreversedString = ""; 

   for (int j = myString.length() - 1; j >= 0; j--) 
   {
      reversed_stringreversedString += myString.charAt(j);
   }

   return reversed_string;reversedString;
}

public static void main(String[] args) 
{
   System.out.print("Insert a 'String': ");

   Scanner input = new Scanner(System.in);
   String userString = input.nextLine().toLowerCase().replace(" ", "");

   if (userString.equals(reverseString(userString))) 
   {
      System.out.println("It is a palindrome.");
   } 
   else 
   {
      System.out.println("It is not a palindrome.");
   }
}