There's plenty of ways/reasons. One is to use it as a side stance. Useful before or after a side kick. It shows up in quite a few of our forms, all the way back to orange belt and all the way up to our 3rd and 4th degree black belt forms (that I've learned so far).
Another option is what we'll do in our self defense training sometimes, which is to move 45 degrees and face our opponent in horse stance.
Then there's sweeps, where you step behind your opponent's leg, make a horse stance in order to get a low center of gravity, and use your knee to execute the sweep.
I know exactly which form you speak, because this comes from another thread.
First off, keep in mind that Taekwondo forms often feature stylized versions of the moves, and the actual application may be somewhat different. (This is something I've really struggled with, especially with forms like Keumgang which feature a lot of cool looking blocks and stances, that I can't find any practical application for the moves as shown and described). So part of the reason it's done this way is to look good.
As to elbow strikes themselves, how I orient my hand will depend on my direction and my level.
- An elbow uppercut or a downward elbow, my palm will face inward towards my ear. I will use the point of my elbow for this one.
- If I'm doing a diagonal elbow (like you see in the UFC when someone is pressed against the wall and wants to score points on the side of someone's head) I would point my palm up. I will try and use the area above my elbow, or the point of my elbow.
- For a cross elbow (elbow starting outside and going inside) my palm will always be down. I will try to use my forearm or the point of my elbow.
- For a straight elbow to the side or rear (if to the front, I will turn so it will be a side elbow), it depends on the level. A lower elbow, such as to the stomach or groin, I will have my palm up. A higher elbow, to the chest or face, I will have my palm down. With both, I am trying to use the area above me elbow or the point of my elbow, depending on how far the target is.
As to my other hand, in each of the above situations:
- My other hand is either in guard, doing another strike in the combination, or holding them in place
- My other hand is probably trapped
- My other hand may grab my wrist to pull the elbow through, hold onto them to pull them into the elbow, or be part of a guard or combination
- My other hand may be pulling them, but more likely is going to push on my fist to push the elbow through
For the "unnamed form", if it is the move I'm thinking of, you are driving your elbow into their solarplexus, ribs, or neck. However, there are other applications of horse stance elbow. One is what I mentioned to Headhunter - put your stepping leg behind them, and then elbow to the chest (even better if you hit the solar plexus) to execute the sweep.
Now, there are other parts of the form I have bigger problems with. For example, the mid-level knife-hand block, followed by a grab and a punch. This was a recent change from a face-level block, grab, and punch, which would make sense (grab the back of their head and punch). I'm not really sure what I'm supposed to grab in that way at mid-height.