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What's the difference between classifying a mathematical statement as a definition and a property? From what I've read so far, the mathematical statements written as property or definition are almost identical. So it confuses me when is a mathematical statement a definition versus a property? And does property need proving?

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    Can you give an example of some of properties you've seen? I guess to me, "property" conjures up things like "Properties of logarithms" like $\log(xy) = \log x + \log y$. – 2017-01-06

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A definition is something the author gets to decide — more or less. There are canonical ways to define many things, but there are sometimes competing and non-equivalent definitions of terms.

For example, if I am writing an analysis book and want to define what an "increasing function" is, I might write that it means that if $x

"Property" is a more vague word. It usually follows from a definition. Furthermore, a property usually describes special classes of things.

For example, one property of increasing functions (on closed intervals) is that they are integrable. We could also state this fact as a theorem.

As another way to think about it, definitions are almost always a way to correlate english words with mathematical statements. We want to know how to interpret the intuitive english phrase "increasing functions" with a rigorous mathematical statement.

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Maybe you mean these two ways to define a set? $$ S = \{ 2,3, 4, 5 \} $$ Here we listed the elements explicitly.

Then we have this kind of definition: $$ S = \{ n \in \mathbb{N} \mid n \bmod 2 = 1 \} $$ This definition of the set $S$ selects elements from $\mathbb{N}$ by the property $n \bmod 2 = 1$ of these elements. Only those elements of $\mathbb{N}$ where that property is true for are part of $S$.

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From Wikipedia:

A definition is defined as:

In mathematics, definitions are generally not used to describe existing terms, but to give meaning to a new term.

A property is defined as:

In math terminology, a property $p$ defined for all elements of a set $X$ is usually defined as a function $p: X \to \{\text {true, false}\}$, that is true whenever the property holds; or equivalently, as the subset of $X$ for which $p$ holds; i.e. the set $\{x| p(x) = \text {true}\}; p$ is its indicator function.