Came across "(Informal)" while reading Analysis I by Tao . What exactly constitutes an example or a definition that is formal ? 


Definition 3.1.1. **(Informal)** We define a set A to be any unordered collection of objects, e.g., {3,8,5,2} is a set. If x is an object, we say that x is an element of A or $x \in A$ if x lies in the collection; otherwise we say that $x \not\in A$. For instance, 3 ∈ {1,2,3,4,5} but $7 \not\in {1,2,3,4,5}$.



What exactly does it mean for a definition/example to be informal in Math?
Informal means it is not precise. In the example you quote, a "set" is defined as an "unordered collection", and its "elements" are "objects", without explaining what is an "unordered collection" and what is an "object". It is not a definition as we understand in mathematics, but simply a connection to other things about which the reader might have some idea already.


In fact, in set theory, we don't define "sets" and the "element" relation. Instead, we state axioms that "sets" and the "element" relation are supposed to satisfy, whatever they might be.


