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Whilst I've never practised an X-stop, I was wondering what the benefits of it are? The move consists of a jump, punctuated by the back foot being placed a head of the front foot, and behind it, forming an X shape.
It seems like it would put you off balance, but it may be good for turning? I don't know.
Looking at this motion, I see a few potential interpretations of this motion.
The X-stance being described is move #36 on this page.
A lot of times in application analysis, we need to look at the previous move and the next move to grasp the context of the application. The X stance/backfist could be read as a setup for a hip throw (yes, TKD has throws), while the turn to the supported block (move #37) could be read as the throw itself. For those unfamiliar with the form, this is a turn from the X-stance at 12 o'clock to a front stance at 9 o'clock. Essentially, you are grabbing the opponent's gi (or dobak), and turning your body 270 degrees to throw him on the ground behind you.
In any kind of application analysis, there are multiple analyses to any one movement, and many of them are correct. One movement can have several meanings.
You should actually be able to land quite on balance in this stance: get your instructor to check how you're placing your feet. Make sure your knees are bent and you're dropping your weight low, but without bending your knees so low that you lose strength in the stance.
I would defer to someone more experienced than me, but I think of the main advantage of the stance is it lets you deliver power forward from the jump while enabling you to become immediately grounded and mobile. From the x-stop, you can drop back into a sparring stance, or easily turn in either direction, or even rotate through 180 to defend against an attacker behind you.
In the palgwe form I linked in an earlier comment http://www.youtube.com/watch?v=MNkeTjs833I I think the point is that the supported-uppercut finishes an opponent, and the student is then ready to answer an attack from any direction: in the form, it comes from the right.
There are a couple of benefits of the x-stance (or x-stop as you call it).
Reach. If you are in a middle or front stance and you simply slide forward, your body weight and torso are centered between your feet (Mostly, there is some percentage difference in weight distribution, but for the purposes of this they are equal). Now, your torso is 6" to 1' or more behind the front foot of your stance. Slide your rear foot up into the X stance, and you gain that foot or so of reach.
Follow up strike. You are pre-tensioning the lateral muscles of your leg for a follow up side kick with your front leg. Because of the angle, your body already wants to release tension and move the front leg forward, so this gives a bit more speed and power to the kick. (And, as mentioned above, you are already much closer and not having to move stances to perform the kick.)
Now admittedly, the techniques will be slightly weaker because you aren't coming from a solidly grounded stance, but the reach may or may not make the x stance worth consideration in a particular situation.
In karate we call that juji-dachi (crossed stance). It is used extensively throughout this kata (https://www.youtube.com/watch?v=HHtm-eJYcg8), where it is either the ending or the beginning boundary for a technique.
Note: You don't stand in the stance and throw techniques, you throw techniques either starting or finishing in the stance. The stance is what happens between techniques.
In the first usage, dropping into the stance allows 180 degrees of movement from the hips and upper body, generating a large amount of force in the elbow strike. It also sets you up for the whiplash strike and block combination that immediately follows.
It is used again almost immediately as the practitioner drops into it to gain distance from the next attacker while receiving an attack, then gets the 180 degrees of hip rotation to generate a powerful punch.
It's next use is almost invisible, but is mirrored in almost every form in any style. When turning in a form, in any style, you pass through the stance at least briefly. This is the source of the power for those impressively powerful blocks in beginner forms.
At about the midpoint in the form, there is an application similar to the one demonstrated in @Rophuine 's comment above. The practitioner moves aggressively sideways into a block or strike, with the intent of immediately exploding in the opposite direction.
Towards the end, the practitioner drops into this stance to perform a jamming technique. The bunkai (analysis) is that it's a stable base that can be dropped into when the intended punch was interrupted by the opponent charging, to execute a jam and set up for the much stronger followup attack.
That is a reinforced-backfist with the left foot crossed behind the right. Aside from the striking application I can't figure out how to directly apply that specific part to self-defense. If you can see the previous part as a takedown the backfist (x-stop) could be a followup strike to put down your opponent. The previous part being the high block/low block/back stance that gets converted to the high block/inside chop/front stance. That can be construed as a release from a rear bear hug or both hands being grabbed from behind. I hope this makes sense. I'll ask my master and respond.
In ITF Taekwon-do, in pattern Yul Gok, there is an x-stance with a high left back-fist side strike; the student gets into the stance by jumping from the previous move. When I learned the pattern, the jump was explained as either jumping over a fallen attacker to strike the next attacker, or jumping a small stream or low wall to close with an attacker. The x-stance helps you stop the motion of the jump in a narrower stance than (for instance) walking stance. It also gives you the flexibility to turn out of the x-stance to perform another technique.
