J, 5 characters and APL, 8 characters

J

10?10

J has the built-in deal operator (?). Thus, we can take 10 out of 10 (10?10).

APL

1-⍨10?10

APL has the same operator which unfortunately starts with one instead of zero. We are therefore subtracting one from each number (1-⍨X means X-1 due to the commute operator).

Ruby, 22

$><<[*0..9].shuffle*''

Python 2.7 (64 63 57)

Not a chance here compared to the operator heavy languages and due to the lack of default loaded random :) This is the shortest I could come up with;

from random import*
print''.join(sample("0123456789",10))

It creates a range and samples 10 numbers from it without replacement.

(Thanks to @xfix for the shorter import format fix and @blkknght for pointing out my somewhat über complicated sampling range)

Python 2.7 (40)

If you run it from the interactive prompt and can read comma separated, you can shave it to 40, but it feels a bit like breaking the spirit of the rules;

from random import*
sample(range(10),10)

GolfScript, 12 characters

10,{;9rand}$

Simply generates the list of digits (10,) and sorts it {...}$ according to some random keys - which yields a random order of the digits.

Examples (try online):

4860972315

0137462985

Perl 6 (18 characters)

say [~] pick *,^10

This generates array containing all random elements (pick *) from 0 to 9 (^10), concatenates them ([~] is reduce concatenation operator), and outputs the result (say).

Sample output:

$ perl6 -e 'say [~] pick *,^10'
4801537269
$ perl6 -e 'say [~] pick *,^10'
1970384265
$ perl6 -e 'say [~] pick *,^10'
3571684902

Octave (14)

randperm(10)-1

randperm unfortunately creates a selection from 1..n, so have to subtract 1 at the end to get 0-9.

PHP - 39 Characters

<?=join('',array_rand(range(0,9),10))?>

I had an 18-character solution that should theoretically work but PHP is weird.

Or, if you want an xkcd answer:

<?="5398421706" // Chosen by program above; guaranteed to be random ?>

EDIT: Thanks xfix, it's now 5 characters shorter and complete. EDIT AGAIN: Live example.

R (23 characters)

cat(sample(0:9),sep="")

Sample output:

> cat(sample(0:9),sep="")
3570984216
> cat(sample(0:9),sep="")
3820791654
> cat(sample(0:9),sep="")
0548697132

In sql server

DECLARE @RandomNo varchar(10)
SET @RandomNo = ''

;WITH num as (
SELECT 0 AS [number]
Union 
select 1
Union 
select 2
Union 
select 3
Union 
select 4
Union 
select 5
Union 
select 6
Union 
select 7
Union 
select 8
Union 
select 9
)
SELECT Top 9 @RandomNo = COALESCE(@RandomNo + '', '') + cast(n.number AS varchar(1))
FROM numbers n
ORDER BY NEWID()

SELECT cast(@RandomNo AS numeric(10,0))

See Demo

OR something similar (courtesy of @manatwork) using recursion and xml.

with c as(select 0i union all select i+1from c where i<9)select i+0from c order by newid()for xml path('')

C# - 145

using System;
using System.Linq;
class P
{
    static void Main()
    {
        Enumerable.Range(0,10).OrderBy(g => Guid.NewGuid()).ToList().ForEach(Console.Write);
    }
}

Condensed:

using System;using System.Linq;class P{static void Main(){Enumerable.Range(0,10).OrderBy(g => Guid.NewGuid()).ToList().ForEach(Console.Write);}}

Mathematica, 27

Row@RandomSample@Range[0,9]

JavaScript, 90 characters

EDIT: Thanks to Rob W, code length is reduced to to 90 characters.

Pretty straightforward way: pick random element of [0,1,2,3,4,5,6,7,8,9] array and append it to the output, then reduce array and replay.

Old version (106 characters):

a=[0,1,2,3,4,5,6,7,8,9],l=11,t="";while(--l){r=Math.floor(Math.random()*l);t+=a[r];a.splice(r,1);}alert(t)

Readable version:

a = [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], l = 10,t = "";
while(l--) {
  r = Math.floor(Math.random() * l);
  t += a[r];
  a.splice(r, 1);
}
alert(t);

New version (90 characters):

a="0123456789".split(t=""),l=11;while(--l)t+=a[r=0|Math.random()*l],a.splice(r,1);alert(t)

JSFiddle: http://jsfiddle.net/gthacoder/ZAJsT/.

JavaScript (80 characters)

alert("0123456789".split("").sort(function(){return .5-Math.random()}).join(""))

JS-Fiddle: http://jsfiddle.net/IQAndreas/3rmza/

K/Kona (6)

-10?10

As with J, ? is the deal operator; the - forces the values to not repeat.

TI-BASIC, 6 bytes

:Disp randIntNoRep(1,10

Mathematica 40

The number is created as a string so as to allow zero to be displayed as the first character, when needed.

""<>RandomSample["0"~CharacterRange~"9"]

Output examples

"0568497231"
"6813029574"

Explanation

"0"~CharacterRange~"9" is infix notation for `CharacterRange["0","9"]". Either of these returns the list, {"0","1","2","3","4","5","6","7","8","9"}.

RandomSample[list] by default returns a permutation of the list. (It can also be used for other kinds of sampling, when parameters are included. E.g. RandomSample[list, 4] will return a Random sample of 4 characters, with no repeats.

PHP, 29 chars

<?=str_shuffle('0123456789');

With PHP, the closing tag isn't required. But if that's against the rules, then you can replace ; with ?> for 1 net increase.

Forth, 72

needs random.fs : r ': '0 do i loop 9 for i 1+ random roll emit next ; r

Room still to golf, maybe, but Forth made this one hard. I think.

Shell/Coreutils, 23

shuf -i0-9|paste -sd ''

F#, 71 characters

I'm new to F#, but here's what I've come up with:

{ 0..9 } 
    |> Seq.sortBy (fun _ -> System.Guid.NewGuid())
    |> Seq.iter (printf "%d")

compacted:

{0..9}|>Seq.sortBy(fun _->System.Guid.NewGuid())|>Seq.iter(printf "%d")

This is not much smaller than JMK's answer, but here's a slightly smaller C# solution (135):

using System;
using System.Linq;
class P { 
    static void Main() 
    { 
        Console.Write(string.Join("", "0123456789".OrderBy(g => Guid.NewGuid()))); 
    } 
}

Compacted (134):

using System;using System.Linq;class P{static void Main(){Console.Write(string.Join("", "0123456789".OrderBy(g => Guid.NewGuid())));}}

Alternate version (135):

using System;
using System.Linq;
class P { 
    static void Main() 
    { 
        "0123456789".OrderBy(g => Guid.NewGuid()).ToList().ForEach(Console.Write); 
    } 
}

Compacted:

using System;using System.Linq;class P{static void Main(){"0123456789".OrderBy(g => Guid.NewGuid()).ToList().ForEach(Console.Write);}}

They're equal in length, but it really just depends on whether you want to use Linq's ForEach function or String's Join function. I was able to remove 10 characters in length by spelling out the range "0123456789" in a string instead of using Enumerable.Range(0, 10).

Prolog, 177/302 characters

I'm a beginner on Prolog, so probably this is not the most condensed code.

:- use_module(library(clpfd)).
sort(N) :-
    N = [N0,N1,N2,N3,N4,N5,N6,N7,N8,N9],
    domain([N0],1,9),
    domain([N1,N2,N3,N4,N5,N6,N7,N8,N9],0,9),
    all_different(N),
    labeling([],N).

Returns:

| ?- sort2(N).                                         
N = [1,0,2,3,4,5,6,7,8,9] ? ;
N = [1,0,2,3,4,5,6,7,9,8] ? ;
N = [1,0,2,3,4,5,6,8,7,9] ? ;
N = [1,0,2,3,4,5,6,8,9,7] ? ;
N = [1,0,2,3,4,5,6,9,7,8] ? 
yes

If you want it to return an integer:

:- use_module(library(clpfd)).
sort(M) :-
    N = [N0,N1,N2,N3,N4,N5,N6,N7,N8,N9],
    domain([N0],1,9),
    domain([N1,N2,N3,N4,N5,N6,N7,N8,N9],0,9),
    all_different(N),
    labeling([],N),
    M is (N0*1000000000)+(N1*100000000)+(N2*10000000)+(N3*1000000)+
         (N4*100000)+(N5*10000)+(N6*1000)+(N7*100)+(N8*10)+N9.

Returns:

| ?- sort(N).
N = 1023456789 ? ;
N = 1023456798 ? ;
N = 1023456879 ? ;
N = 1023456897 ? ;
N = 1023456978 ? 
yes

Using instead:

labeling([down],N)

Gives the numbers in the opposite order:

| ?- sort(N).                                        
N = 9876543210 ? n
N = 9876543201 ? n
N = 9876543120 ? n
N = 9876543102 ? n
N = 9876543021 ? 
yes

Unlike some other codes posted, this returns all possibilities (with no repetitions).

Scala, 37

util.Random.shuffle(0 to 9).mkString

LOGO, 64 characters

make "d 1234567890
repeat 10 [
    make "n pick d
    show n
    make "d butmember n d
]

pick returns random item of the supplied list. butmember returns list with all occurrences of the specified item removed. Note: Not all Logo implementations support butmember command.

Racket 45 43

(map print(shuffle'(0 1 2 3 4 5 6 7 8 9)))

php

function gen_uid() {
$random_id_length = 10;
$rnd_id = crypt(uniqid(rand(),1));
$rnd_id = strip_tags(stripslashes($rnd_id));
$rnd_id = str_replace(".","",$rnd_id);
$rnd_id = strrev(str_replace("/","",$rnd_id));
$rnd_id = substr($rnd_id,0,$random_id_length);
$rnd_id = strtoupper($rnd_id);
return $rnd_id;
}
$file_uid = gen_uid();

this will make you get 10 characters like tah

1UOOPDIQBA

you can increase the number of charckrs by changeing his value

$random_id_length = 10;