We can say a lot about infinity. But for people who don't study advanced math by choice, the first question is why are we talking about infinity, when we don't normally, in our physical world, encounter infinite amounts of anything.

The answer is pretty simple: it is easier to work with infinity than with a large finite number, and by working with infinity, you discover similarities among situations in which you encounter different "large finite numbers". So, for example, it is easier to study a circle than a polygon with a million sides. In statistical physics we say that the heat capacity "diverges" at a phase transition, but of course a real material must absorb a finite amount of heat at a phase transition; nonetheless, we treat "macroscopic" as "infinite" freely, knowing that any errors will be far below measurement limitations.

In the above cases, we are dealing with infinity as the limit of a particular process. But in order to make the ideas in calculus convenient, we want to consider all of the limits of all sequences which converge, because it allows us to use the concept of limit freely. This leads to the definition of the "real numbers" by Dedekind cuts (the topological closure of the rationals); it is how we define the "bigger infinity". When we say the real numbers are "bigger" than the integers, we can explain this in finite-sounding terms as: we cannot define a "sequence of all sequences".