You would think there was just one infinity but it turns there are quite a few different kinds that have different sorts of mathematical properties or behaviours. 

The first most common classification of infinities  is whether it is countable or uncountable. 

Cardinal numbers are often the first example of a set of natural numbers that can be infinite but countable. 

Aleph null, the smallest infinite cardinal


The dots mean going on for infinity.

Ordinal numbers ( this very roughly means they have an order from say high values to low values)  also make up another kind of set of infinite numbers, these have the symbol lower case omega ω . They start with the natural numbers, 0, 1, 2, 3, 4, 5, … After all natural numbers comes the first infinite ordinal, ω, and after that come ω+1, ω+2, ω+3, and so on. 

A graphical “matchstick” representation of the ordinal ω². Each stick corresponds to an ordinal of the form ω·m+n where m and n are natural numbers. image: wikipedia


See also a list of different infinities by