When a person faints, they remain unconcious. The persons fainting can be unknown so placing them in certain positions such as on their side will prevent choking in the person vomits. Also, bringing the person legs to their chest can allow easier blood flow through the heart, allowing oxygen to flow throughout the body better and faster. Sometimes, laying the person flat is recommended because the head and heart remain at the same level.
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Do this by putting the victim on their back (if they do not have any other injuries). Tip their head back slightly, and listen for 5 seconds for breathing.
Then check their heart rate, using two fingers. Put your pointer and middle finger on their wrist and count to 10.
If Victim is not breathing or no pulse (or other major injuries)...
If Victim is breathing...
I would suggest to become certified in General First Aidand CPR/AED Adult and Children with the Red Cross. They will more in depth go through video procedures and fully instruct you step by step. I have given you a few hints, but the full course cannot be explained in this answer.
For extended information of what to do in other situations:
go to www.redcross.org
I would definately try to catch them if possible to break their fall and prevent a head injury. Keep them still and don't try to move them until it has been determined that it is safe to move them or they are in danger where they have passed out.
Or...
lay them on a flat surface on their back and elevate their feet at least 6 inches so the blood could flow back to their head