Question: I live in San Antonio. Last week, I hired these guys to trim my trees. As they worked, they were joking around with each other and one guy kept saying (as he hung from a branch), "¡Auxilio! ¡Auxilio!" Well, I had never heard that before, so I searched and found auxiliar in the dictionary — it means to assist, or render aid, so I assume that auxilio is a cry for help, though why wouldn't it be auxíliame? Is this a more common usage (among those from Mexico) than "¡ayúdame!"?
Answer: You're probably thinking of auxilio as conjugated verb form (which it can be); instead, in this case, think of auxilio as a noun meaning "help." So the man was jokingly asking for help rather than saying the same thing in verb form. The distinction doesn't translate directly into English; suffice it to say you don't have to remember how to use a verb in imperative form if you yell for help.
Yes, you can say ayúdame (or more formally, ayúdeme or ayúdenme, depending on how many people you're talking to) as a way of requesting help. In fact, that verb form is quite common, but it doesn't convey the sense of urgency that "¡auxilio!" does. It's somewhat like the difference between "help!" and "assist me."
The noun socorro also could be used instead of auxilio. Either word would be understood everywhere Spanish is spoken; it's a matter of regional usage or personal choice which one you should use.