Glad I could help explain.

First, the "division by zero" issue. I perhaps should have clarified this point. Never is an actual division by zero appropriate. However, there are situations in which we can nearly do so. This is closely related to the notion of "Limits". If you've had a calculus course, you're undoubtedly familiar with them. In the event you haven't, I'll provide a very simplified ( and not-quite correct ) description. Think of a limit as an action which analyzes the behaviour of a function ( or sequence ) as you get "close" to a point. Consider the functions f(x) = sin( 1/x ) and g(x) = x*f(x) = x*sin( 1/x ) for small x ( when I say small, I mean small in magnitude ). As x gets really close to 0, 1/x begins to "cover the real line" at a very rapid pace. In fact, 1/x covers all numbers larger than 1 "at the same rate" as y = x covers the numbers between 0 and 1. To conceptualize the graph of f(x), think of the sine curve, and take the entire graph of sin(x) on the right side of the vertical line x = 1, and cram that 'backwards' into the space between x=0 and x=1. This is precisely what f(x)=sin( 1/x ) looks like near 0. From this, it is apparent that f(x) oscillates back and forth from -1 to 1 infinitely often between 0 and any positive number. This is reinforcement for the fact that 1/0 is bad mojo, it makes seemingly normal functions behave VERY badly. However, consider g(x) = x*f(x). As x gets close to 0, g(x) will still oscillate infinitely many times, but now, instead of going between -1 and 1, it will oscillate between -x and x. Therefore, as x gets smaller, so does g(x). This is not the case with f(x). Essentially, what we are saying here is:
Note that in neither the case of f(x) or g(x) do we consider the value at x = 0 to have any meaning. In fact, we simply say "f and g are not defined at 0". However, while f doesn't seem to follow any regular behavior close to zero, g(x) is getting really small. And this is what a limit is. As x approaches 0, g(x) also gets close to 0. Therefore, we'd say: "The limit as x goes to 0 of g(x) is 0"
So we didn't divide by 0 ( in the sin(1/x) ) but in the case of g(x), we were able to consider the way the function behaves at 0 ( I italicize at because the precise concept requires a bit more detail than is necessary here ).
There are also other situations where 'division by zero' seemingly pops up. This is in a field called complex analysis where there are special classes of functions defined on regions of the complex plane. The functions have some very nice properties, namely, the ability to evaluate these functions by looking at the singularities of other functions ( a singularity is a place at which a function is not defined, ie 1/z has a singularity at z = 0 ). Specifically, if f(z) is one of these special functions ( called holomorphic ), then if I know the value of f(z) on every point of a circle in the complex plane, and f(z) is defined everywhere in that circle ( accurately speaking, I should say "Holomorphic in the circle", but that's not very helpful ), then I can determine the value of f(p) for any p inside the circle simply by looking at the function g(z) = f(z)/(z-p) on that circle! Notice how g(z) isn't even defined at z = p, but somehow, it tells me everything I need to know about f at the point p. So while actually dividing by zero is bad, looking at cases where you're almost dividing by zero can be very very good.

Now, regarding infinity. Again, limits pop up. First, a caution: Whenever someone says "infinity", unless it's very clear what they are talking about, you should seek clarification. Infinity is a term that gets thrown around a lot, but has very different meanings in very different circumstances. As an example of this, we'd say that the size of the rational numbers are infinite, and the size of the real numbers is infinite. But they are not the same size!! The reals are, in a way, infinitely bigger than than the rationals. Additionally, in an example I'm about to give, you'll see me use the terms "negative infinity", "positive infinity" and "infinity" ( -inf, +inf, and inf for short ), but they will all represent different things, some you might be familiar with. In any case, if you take to heart anything I've written, remember this: Infinity is, unfortunately, an overused and underdefined term, so make sure you know what "infinity" someone refers to when they use it. Anyway ....
Consider the real numbers, aka "The Real Line", 1 Dimensional Euclidean Space, or R.
When one says "positive infinity", with regards to R, one usually means: The limit of f(x)=x as x increases without bound", or in other words, something that is greater than any real number. When someone says "negative infinity" with regard to R, one usually means: "The limit of f(x) as x decreases without bound", or something that is less than any real number. It is crucial at this point to point at that +inf and -inf are NOT part of the reals, they are instead an abstract concept which is used mostly for notational purposes. They allow us to say things like, "The limit as x goes to -inf of e^x is 0" However, there is a notion of infinity which possesses an important value when combined with the real numbers.
Consider the function f(x) = 1/x^2. As x gets close to 0, whether we approach from the positive or negative side of 0, f(x) gets bigger and bigger and bigger. In fact, it increases without bound. Similarly, g(x) = -1/x^2 as x approaches 0, g(x) gets more and more negative, decreasing without bound. Both f and g behave very nicely for every real number with the exception of x = 0. As x approaches 1, g(x) approaches g(1) = -1. As x approaches 10, f(x) approaches f(10) = 0.01. This is a good thing ( namely, we call this "continuous" ). However, near 0 things seem to break down. This is where "infinity" comes in. Thru a process called "compactification" ( compactification means "to make compact" and compact is a property of a (topological) space which basically describes how tightly packed a space is ). Since the real numbers have no greatest or least element ( recall +inf and -inf not in R ), it's not 'compact', we can't really grab all the real numbers at once ( I'm abusing the language a great deal to avoid describing the concept of compactness in more detail ) . However, if we include a point "At Infinity", then we can grab a bunch of numbers around, say 0, and a bunch of numbers "Around Infinity" and we'll get all the reals! This is compactification.
To better visualize what is happening: Imagine if you will , the real line drawn vertically in the plane ( say the line x = 1. ) Additionally, imagine the unit circle with the point (-1,0) removed. Now, try to envision the line being wrapped towards the circle, but shrinking as it bends, such that when you're done wrapping it:
This is essentially the inverse of a Stereographic Projection. Now, simply add 1 point to the plane, at (-1,0) and call it "Infinity". That's essentially what's going on; you're gluing the ends of the real numbers to a point outside the real numbers. So, we have the "Compactification of the Reals", which is commonly known as "The ( projectively ) extended real number system", and denoted by R*. So why do we care? Because now, in R*, functions like f(x) = 1/x^2 and g(x) = -1/x^2, while not defined at 0, no longer behave badly at 0, because they approach "Infinity". Typically, what one might do in this case is define a compound function such as, "f(x) = 1/x^2 for x not 0, and f(0) = 'Infinity' " ( note this is in R* not R ). You might be thinking: "but -1/x^2 goes to negative infinity!", but this is not the case in R*, and this goes back to what I was saying earlier about infinity. However, if you wanted to, you could compactify the reals by adding the points "-Inf" and "+Inf" to the "Left and Right ends of the Real Line". This also is compactification of the real line, and it is called "The (Affinely) Extended Real Numbers" and is denoted R-bar.

Anyway, I'm sure this is way more than you wanted to know, and I apologize for the simplistic manner in which I tried to present everything, but brevity was the goal. When accuracy was sacrificed -and it was painful to do so- , I tried my best to put down a note indicating the exception. If you have any more questions, I'd love to try and answer them, but I'll need to be more brief next time around )

-joeboo

ps - I happened to notice some questions you posted in this thread regarding the center of the universe, and this provides a good light to view it in. Go back to that business with the circle and the line in the plane. Remember when you had wrapped the line around the circle, prior to adding in "Infinity"? Well, in an effort to explain why there is no center, consider the following: You can see the entire circle, but remember it's really the real line. Even though 0 is conveniently in between the positive and negative reals, realze that -1000 is just as conveniently placed in between all numbers less than -10000 and all numbers greater than -10000. Now, when you consider this on the "punctured circle", you see that even though 0 looks like a good place for a center, as soon as you "Compactify the reals" by adding the Infinity in, it becomes even more apparent that the decision to call 0 the center is arbitrary. Also, you can't consider the point (0,0) the center, because there is no such analogue in the real numbers. The fact that you see it at all is because we "embedded the reals into the plane" ( or in more general terms, we embedded the space into a higher dimensional space ). Finally, even tho you can see the size of the punctured circle, ( and measure it's circumference from a higher dimension ), it is still infinite. If you don't see this, or want to know why this "embedding nonsense" is important ( or what it even means ) I'll try to explain it. In any case ... it's time to sleep =/

( edit: typo )