Sounds like you need an expression like this:

^[0-9a-zA-Z]+(,[0-9a-zA-Z]+)*$

Posix allows for the more self-descriptive version:

^[[:alnum:]]+(,[[:alnum:]]+)*$
^[[:alnum:]]+([[:space:]]*,[[:space:]]*[[:alnum:]]+)*$  // allow whitespace

If you're willing to admit underscores, too, search for entire words (\w+):

^\w+(,\w+)*  // allow whitespaces around the comma

You can give a name to your big regex in PCRE like:

(?<big>[a-zA-Z0-9]+)

Everything after the ?<name> will be recorded with the name given.
Called Regular Expression Subroutines

So, repeats (?&name) become easy:

^(?<big>[a-zA-Z0-9]+)(,(?&big))*$

Test it Online

So, matching an IP, for example, becomes simpler:

^(?<ip>25[0-5]|2[0-4][0-9]|[01]?[0-9][0-9]?)(.(?&ip)){3}$

Test it OnLine.

Use it with grep as:

grep -P '^(?<ip>25[0-5]|2[0-4][0-9]|[01]?[0-9][0-9]?)(.(?&ip)){3}$'

You can simply replace the middle number of each line with nothing.

In the editor

That is, in your editor, search-and-replace the regex

,[0-9]+,

(which only matches numbers with commas on both sides, which for your input is just the middle number) with a single comma:

,

I assume Ultraedit supports regex search-and-replace. If not, try Notepad++, which I know does.

From the command line

Since you tagged your question shell-script, here's how to do it from the command-line.

sed

Use sed, a standard Linux command also available for Windows as part of Cygwin or GnuWin32:

C:\>sed -e 's/,[0-9]+,/,/g' filename.txt

Powershell

Jens pointed out that you can also do it in Windows Powershell; see this explanation.

1. Both string1 and string2 can contain only alphabets.

[a-zA-Z]

The - is used to represent a range in character classes. Here we say that, it can be any alphabet between a and z or A and Z.

2. Both string1 and string2 length shall not be more than 20
3. string1 and string2 can not be empty

[a-zA-Z]{1,20}

It means that, match minimum 1 and maximum 20 alphabets.

4. only 2 string can be present (string1,string2,string shall be invalid)

/^[a-zA-Z]{1,20},[a-zA-Z]{1,20}$/

^ represents the beginning of string and $ represents the end of the string.

You can check that the RegEx works fine for all your inputs mentioned in the question, like this

var invalid_cases = ['Edward,  john', '12*&,john', 'Edward,12#', ',         ',
        'eeeeeeeeeeeeeeeeeeeee,jjjjjjjjjjjjjjjjjjjjj'],
    valid_cases = ['Edward,john'],
    regEx = /^[a-zA-Z]{1,20},[a-zA-Z]{1,20}$/;

valid_cases.forEach(function(currentString) {
    console.assert(regEx.exec(currentString)[0] === currentString);
});

invalid_cases.forEach(function(currentString) {
    console.assert(regEx.exec(currentString) === null);
});

I suggest you to do in the following way:

(\d+)(,\s*\d+)*

which would work for a list containing 1 or more elements.