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Well, I'd have figured my definition would be in here already, but it's not, so I'll take a stab at this.

The Value

The best definition for infinity that I know of is

             n
inf = lim   ___ 
      x->0   
            |x|

where n is any positive real number, and |x| approaches downward to zero (as noted by Professor Pi). I prefer to actually set n to 1 when I define infinity, because certain math problems arise where the limits of two functions f(x) and g(x) both approach infinity, but for any x g(x) is not the same value as f(x). For example, g(x) might be defined as twice f(x). In these cases, the limit of f(x) would be infinity, and the limit of g(x) would be 2 · infinity. To me, defining infinity as the limit of 1 divided by x as x approaches zero is easiest on my brain as it clears any ambiguity. I just use the above definition with n=1 and I know how large any infinite value is relative to any other. When you factor in the fact that you can have negative infinite values, well, I think it just makes sense.

Personally, I do think this definition makes the most sense because it can rearranged as a definition of zero:

                   n
zero =  lim       ___
       x->∞    
                   x

In my mind, n in this case is not so important since any number times zero is zero, but still, I prefer n to be one (at least, when dealing w/ limits). So there you have it. Well, my take on it at least.

The Symbol

The proper name for the symbol is actually infinity and it looks like an 8 rotated 90°. Supposedly, some browsers (supporting HTML 4.0) should be able to display infinity as ∞ or ∞ (but mine doesn't). In case yours does, the HTML for it is (or should be) ∞ or ∞ To use the former, your browser must support HTML 4. Ability to use the latter of course depends upon your system being able to display the ISO 10646 character set. My browser/OS combination doesn't allow either, but I've included it just in case yours does.

As for the name of the symbol, it is simply "infinity." It has no other fancy name, and in fact has no "official" name. Some do refer to it as lazy eight, and some call it lemniscate. As far as lemniscate goes, it is actually the name of a mathematical system that just looks like the symbol for infinity. The symbol itself was introduced by mathmetician John Wallis in 1655 when he wrote "De sectionibus conicis." The symbol was supposedly borrowed from a Roman symbol for 1 000 and was declared to represent infinity.

Now Txikwa tells me (in addition to the correct spelling of Wallis' treatise on conic sections) "Probably the thousand symbol mentioned is not ordinary M but an old alternative way of writing it: CID where the D is a backwards C. (And half of that gave normal D for 500.)" Sounds reasonable to me.


References
Ken "Dr." Math. Ask Dr. Math - Infinity Symbol. The Math Forum. <http://forum.swarthmore.edu/dr.math/problems/lazy8.html>
W3C. Character entity references in HTML 4. <http://www.w3.org/TR/REC-html40/sgml/entities.html>