The lemniscate symbol (∞, U+221E) is used to represent infinity, a concept that is larger than any natural number.
Set Theory
Cardinality
The sizes of infinite sets are measured using cardinal numbers, using generalised natural numbers. Cardinality is measured using bijective functions, where two sets are equal in size (cardinality) iff they there is a bijective function that can map all elements from set A to set B. Bijection is a function where there is a one to one correspondance between two sets.
Through this, infinite sets can have different cardinalities. The classic example is that the cardinality of the set of real numbers (ℝ) is greater than the cardinality of the set of natural numbers (ℕ), as there is no possible bijective function that can map the set of natural numbers to the set of real numbers.
Cardinalities of infinite sets are represented using aleph numbers, written as ℵ with a subscript denoting the relative size. The cardinality of ℕ that is the smallest infinity cardinality, which is represented by ℵ₀ (aleph-null). The cardinality of ℝ is called the "cardinality of the continuum" and is represented by 2^ℵ₀. ℝ has the same cardinality as the the Power Set of ℕ.
The set of all functions from ℝ to ℝ has an even greater cardinality than that of ℝ, (2^ℵ₀)^(2^ℵ₀).
Countability
Infinite sets can be broken into two kinds, countable and uncountable. There are a number of equivalent definitions of uncountable sets, but the simplest is an uncountable set has a cardinality that is not finite and not equal to ℵ₀.
Transinfinity
Transinfinite numbers are numbers that are larger than the set of all finite numbers, but aren't necessarily absolutely infinite. Absolute infinity is an extension of infinity proposed by Georg Cantor that is a number that is bigger than any other conceivable or inconceivable quantity, finite or transfinite. Cantor associated the Absolute Infinite with God, and it is of course distinct from actual infinity.
Actual Infinity
The acceptance of infinity entities as given, actual and completed objects. This concept is as much a philosophical one as it is a mathematical one, an again is distinct from potential infinity. The hilariously titled "pre-modern thinkers" generally disagreed with the concept of completed infinity, as exemplified in the motto Infinitum actu non datur meaning "there is only potential infinity but not an actual infinity". Wikipedia alleges (without source) that the concept of actual infinity is now commonly accepted.
Potential Infinity
The result of a non-terminating process that could theoretically infinitely produce a sequence of individually finite elements without having a final element itself. This is formalised using the concept of limits with are integral to calculus.