'Infinity' can mean a number of different things. A very basic use is in the area of induction, which gives rise to the natural numbers (1,2,3,4,....). Induction is an axiom that basically states, 'given x, you can get x+1'. The natural numbers then use two basic axioms: '1 exists' and 'induction'. In this context, infinity is then simply the statement that the inductive process can be repeated indefinitely.
[If I recall correctly, 0 is not part of this scheme, and would require its own axiom; however, my recollection on this one is a bit hazy - its a slightly different point, but goes to the actual definition of 0 as an additive identity, with 1 being a multiplicative identity. Philosophically speaking, '1 exists' is also a much easier axiom than '0 exists'.].
A more rigorous statement to that effect is that for any given number, X, the above inductive process will eventually generate a number Y such that Y>X. Since X can be arbitrarily large, the natural numbers are infinite.
Generalising, 'Infinity' is shorthand for saying that the answer to a given process* is arbitrarily large. Or, alternatively: for any given number, no matter how large, at some stage the process will generate an answer that exceeds the number.
*process in this sense is usually a countable series expansion. An example of a countable series expansion is 1+1/2+1/4+1/8+1/16+.... Some countable expansions 'converge' [i.e. the answer is finite, such as the one given], whereas others do not converge and therefore become arbitrarily large. 'Countable' means that a 1 to 1 mapping exists between a given set and the set of Natural numbers, {1,2,3,...}. It is possible to have 'Uncountable' sets, but the explanation of those would be very tangential to this post
When considering 1/0, its possible to use a limit process to determine that the answer must be arbitrarily large. This can be done by substiting a (small) value y into the expression, which becomes 1/y. For any given X, it is then possible to find a value of y such that 1/y > X. In other words, the expression 1/y tends to infinity as y tends to 0, often shortened [technically incorrectly] to 1/0 = infinity.
The final part to this fallacy is then to say 1/0=infinity; therefore infinity x 0 is equivalent to 1/0 x 0. Cancelling the denominator with the numerator, you get 1/0 x 0 = 1.
The problem here is that the statement 1/0 x 0 is actually meaningless. Sure, you could use the limit process above and say that 1/y x y = 1 for any y, no matter how small (i.e. let y tend to 0) but the problem is that 0 is not well-defined in this case; I could just as easily say that 1/0 x 0 is equivalent to 1/y x 2y as y tends to 0. 2y tends to 0 just as y does, and the answer here would then be 1/0 x 0 = 2 [or any other number you care to choose]. Note also, that if I don't use a limit process for the 0 multiplier, then I get a result of 1/0 x 0 = 0.
