I've read explanations online and I can parrot the explanation that it is numbers taken together with operations that can be performed on them and the rules they follow like associativity, commutative etc. But I don't exactly know what that means or looks like. For example set is a collection of numbers so I can write a set $A = \{1,2,3\}$. If I wanted a field for this set do I write $AF = \{1,2,3,1+2,2+3,1+3,1*2,2*3,1*3\}$?
Please explain this as simply as you can without talking in an abstract manner. Concrete examples would be extremely helpful.
I think it's helpful to think in terms of information. If I have a set and I want to tell you what it is, I just have to tell you its elements; then you completely understand what the set is. With a field, however, I need to do more than just tell you what the elements of the field are - I also have to tell you how they add and multiply.
For example, consider a two-element set $\{a,b\}$. There are two distinct fields which use this as their underlying set: we could have $a$ play the role of $0$ and $b$ play the role of $1$, or $a$ play the role of $1$ and $b$ play the role of $0$. The point is:
You should really think of a field as a tuple: a field consists of
a set $F$,
two distinct elements $\underline{0},\underline{1}$ of $F$ (so in particular $F$ has to have at least $2$ elements), and
two binary operations $\oplus,\otimes$ on $F$
such that [various properties]. The set $F$ on its own doesn't tell us too much; to understand the field we need to understand the whole tuple $$\langle F, \underline{0}, \underline{1},\oplus,\otimes\rangle.$$