Remove extra spaces from a string without using Regex or any internal library methods

A few things to point out.

char[] longChars = longString.ToCharArray();

strings in C# are array-like, and can be indexed directly. We don't need to convert it into a char[].

int spaceCount = 0;

We don't really need to keep track of how many spaces so much as whether the last character was a space. Therefore, this can become a boolean flag instead.

if (longChars[i] == 32)

This line has a magic number (32), and that makes it a little harder to understand. I'm guessing it's the Unicode number for the space character, but the fact that I have to guess makes it bad. Better would be: if(longChars[i] == ' '), but best would be: if(Char.IsWhiteSpace(longChars[i])). That catches all potential whitespace characters, and will be more robust. Also, it makes your intention much clearer.

if (spaceCount > 1 && sb.Length > 0)
    sb.Append(" ");

I always recommend using braces in light of goto fail.

//Reset the space count
spaceCount = 1;

This confuses me. You say you are "resetting" the space count, but the original value was 0, not 1, so I might (incorrectly) think this is a bug and try to fix it. The root of this issue lies in how you are tracking spaces.

So incorporating those suggestions, here's what I come up with:

public static string Despace(string longString) {
    StringBuilder sb = new StringBuilder();
    bool lastWasSpace = true; // True to eliminate leading spaces

    for(int i = 0; i < longString.Length; i++) {
        if(Char.IsWhiteSpace(longString[i]) && lastWasSpace) {
            continue;
        }

        lastWasSpace = Char.IsWhiteSpace(longString[i]);

        sb.Append(longString[i]);
    }

    // The last character might be a space
    if(Char.IsWhiteSpace(sb[sb.Length - 1])) {
        sb.Remove(sb.Length - 1, 1);
    }

    return sb.ToString();
}

Use split(), and then go over the String array you got. If some variable is not " ", it's a word like you need.

public class RemoveExtraSpacesFromString {

    public static void main(String[] args) {

        String str = " Today it is going to rain. ";        
        String strResult = "Today it is going to rain.";

        if (compactStringPerfect(str).equals(strResult)){
            System.out.println("Good Job!");
        }
    }

    private static String compactStringPerfect(String spacesStr){

        String newStr = "";
        String currentStr = "";

        for (int i = 0; i < spacesStr.length(); /*no incrementation*/) {
            if (spacesStr.charAt(i) != ' '){
                while (spacesStr.charAt(i) != ' '){
                    currentStr += spacesStr.charAt(i); 
                    i++;
                }               
            }
            else {
                i++;
            }
            if (!currentStr.equals("")){
                newStr += currentStr + " ";
                currentStr = "";
            }
        }

        return newStr.trim();

    }

}

You can use regex to remove extra spaces. In Java, it would look something like this:

public static void main(String[] args) {
    String str = "      Today   it  is     going    to  rain.     ";
    String strResult = str.replaceAll("\\s+", " ").trim();

    System.out.println(strResult);
}

Output:

Today it is going to rain.

Thanks all for sharing your insight. I think using regex is the fastest and optimal solution to this. I believe this was an interview question just to see how one codes and things if no helper functions were to be used.

Another simple way of doing this is

...
char[] WhiteSpace = new char[] { ' ' };
string longString = "   Today   it  is     going    to  rain.     ";
string[] split = longString.Split(WhiteSpace, StringSplitOptions.RemoveEmptyEntries);
string compactedString = string.Join(" ", split);
...