That's quite easy. Just ask Perl to tell you what it sees:

Q:\>perl -w -MO=Deparse -e "s;;new;;old;;g"
Unquoted string "old" may clash with future reserved word at -e line 1
+.
Unquoted string "g" may clash with future reserved word at -e line 1.
Useless use of a constant in void context at -e line 1.
Useless use of a constant in void context at -e line 1.
BEGIN { $^W = 1; }
s//new/;
'???';
'???';
-e syntax OK
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So basically, your program is equivalent to $one =~ s//new/;, which replaces the first empty place in the string $one with new, which is what you see.

Also, this shows again why running with warnings enabled reveals many problematic spots in the code.

As is often the case, adding "use strict" and "use warnings" gives a clue:

Bareword "old" not allowed while "strict subs" in use at ./re line 7.
Bareword "g" not allowed while "strict subs" in use at ./re line 7.
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B::Deparse is another useful tool.

$ perl -MO=Deparse re
my $one = 'TestnewTest';
$one =~ s//new/;
'???';
'???';
print "$one\n";
re syntax OK
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The delimiters for the substitution operator can only be single characters[1]. Your code is therefore being interpreted as:

$one =~ s//new/; 'old';; 'g';
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So the first empty string Perl finds (at the start of $one) is changed to "new".

[1] I feel sure that someone will be along shortly to point out exceptions to that rule :-)

jesuashok how many times do we have to tell you! Always, for every Perl script you write, each time you start writing include:

use strict;
use warnings;
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For the code fragment that you give Perl reports:

Bareword "old" not allowed while "strict subs" in use at noname.pl lin
+e 5.
Bareword "g" not allowed while "strict subs" in use at noname.pl line 
+5.
Execution of noname.pl aborted due to compilation errors.
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Why do I repeat this when several other people have already said the same thing in this thread? Because you didn't hear and heed the other dozen times you have been told in the past! So, just in case you missed it - always use strictures. Capiche? Comprende? Understand?