What does it mean to show that something is well defined?

This is coming from my first course in undergraduate analysis, and it's confusing to me how to show that some operation is "well-defined". For example, my professor left as something for us to figure out on our own, not homework, to show ourselves that if $a,b,c$ and $d$ are integers, and ($b,d\not=0$) that $$\left[\left(\frac{a}{b}\right)\right]+\left[\left(\frac{c}{d}\right)\right]=\left[\left(\frac{ad+bc}{bd}\right)\right]$$ is well defined. He then made an example that said: If $\frac{a'}{b'}\sim\frac{a}{b}$ and $\frac{c'}{d'}\sim \frac{c}{d}$, then $\frac{a'c'}{b'd'}\sim \frac{ac}{bd}$.

Also above in the brackets are supposed to be $2$ classes of element's.

It usually means that the result doesn't depend on some more or less arbitrary choices you have to make.

One example is when you're dealing with equivalence relations (as in your case). The definition of a sum of two equivalence classes is made by choosing representatives for each class. To make this work well, you want the result to be independent of how you choose those representatives.

One other example is if you define a curve integral of a vector field using a parametrization of the curve. In that case you want to show that it doesn't matter which parametriztion you choose; you will still end up with the same value of the integral.

To show that addition is well-defined there means to show that using different elements to represent the same equivalence class leads to the same result. You need to show that if $[\frac{a}{b}]=[\frac{a'}{b'}]$ and $[\frac{c}{d}]=[\frac{c'}{d'}]$, then $[\frac{ad+bc}{bd}]=[\frac{a'd'+b'c'}{b'd'}]$.

In mathematics it is often the case that you have an interesting set $S$ of objects (the set ${\Bbb Q}$, the set of all curves of some sort, etc.), but the individual elements of this set do not have a unique presentation. Instead you have a much larger "set of representants" $\tilde S$ and a "production scheme" $\phi$ that for each representant $x\in\tilde S$ produces a unique element $\phi(x)\in S$.

Now you want to define (a) a property $P$ that some elements of $S$ might have and others don't, or (b) a function $f:\ S\to X$ from $S$ to some set $X$; but the only way to address the individual elements of $S$ is by means of their representants $x\in \tilde S$ and the scheme $\phi$.

The way to go about this is in case (a) that you define a property $\tilde P$ for the elements $x\in\tilde S$ that reflects your intentions. Then you prove that the element $\phi(x)\in S$ possesses the intended property $P$ iff $x\in\tilde S$ possesses property $\tilde P$. Thereafter you can say that property $P$ is well-defined on $S$.

Similarly in case (b): Here you define a function $\tilde f:\ \tilde S\to X$ and then have to prove that $\tilde f(x)=\tilde f(x')$ whenever $\phi(x)=\phi(x')$. Thereafter you can say that a "well-defined" function $f:\ S\to X$ has been established. It is related to $\tilde f$ and $\phi$ via $f\circ\phi=\tilde f$.