I've seen multiple definitions of cantor sets but they are all look different than mine. My book defines a cantor set as:
The set of all real numbers of the form $\sum_{n=1}^{\infty}a_{n}3^{-n}$ where $a_{n}$ takes one or other of the values $0$ or $2$.
How is this a set? I don't understand what they mean with "where $a_{n}$ takes one or other of the values $0$ or $2$" does it mean that $a_{n}$ alternate like $0$, $2$, $0$, $2$? Could you guys give me some values in this set? And what does it have to do with this image that i see everywhere?
Your book means that the cantor set is the set of numbers $x$ that are possible to write in the form $\sum_{n=1}^{\infty}a_n3^{-n}$ for some sequence $a_n$ where each $a_n$ is either $0$ or $2$. A bit less densely, you might say either:
A number in $[0,1]$ is in the Cantor set if it can be written as twice the sum of distinct powers of $3$.
A number $x$ in $[0,1]$ is in the Cantor set if it has a ternary expansion that never uses a $1$. (This is the same as above, realizing that ternary expansions are just "write a decimal point then a bunch of numbers $\{0,1,2\}$ and consider the sum of the $n^{th}$ term times $3^{-n}$ over all $n$")
The particular $x$ where $a_n$ alternates between $0$ and $2$ is therefore in the Cantor set (this $x$ equalling $1/4$), but there are uncountably many other sequences $a_n$ whose only values are $0$ and $2$, all of which yield distinct elements of the Cantor set.
The image you show shows constructing the same set by taking an interval and repeatedly removing the middle third of each interval. This yields a sequence of sets that get smaller and smaller - and the intersection of all of those sets is the cantor set, and is exactly the same set your book defines. The equivalence is most clear in ternary expansions:
At first, you have the interval $[0,1]$. You then remove the interval $(1/3,2/3)$ because the first term of their ternary expansion must be $.1\ldots_3$, meaning they cannot be written in the desired form. Then, you remove $(1/9,2/9)$ and $(7/9,8/9)$ whose ternary expansions start $.01\ldots_3$ and $.21\ldots_3$ because, while their first digit is okay (being $0$ or $2$), their second digit is not. You would then remove those numbers whose ternary expansions begin $.001\ldots_3$ or $.021\ldots_3$ or $.201\ldots_3$ or $.221\ldots_3$ and so on - and the only numbers left at the end would be those that can be written with a ternary expansion containing only $0$'s and $2$'s - which is exactly the set of numbers that can be written in the form your book posits.