Will has already said many of the things I would have said trying to answer your extended question, but let me add a few things without trying to avoid overlap.
In fact, let me start with an overlap: Indeed, a fundamental question of birational geometry is what you asked: "What are some interesting properties of varieties that are preserved under birational transforms?" Studying this question is already interesting, but let me say a bit more.
Algebra vs. Geometry I know this is the original question, but I would like to record my favorite comment about this issue. I would actually say that (at least with respect to algebraic geometry, and hence to birational geometry), Algebra=Geometry!
Let me give two quick examples:
a) Resolution of singularities of curves, a.k.a., starting with an arbitrary (=possibly singular, non-compact) curve and constructing a smooth projective curve which is birational to the original one. You might agree that this is a "purely" geometric question. However, one way to do it is to consider the set of DVRs of its function field (plus some technical condition), define first a topology, then regular and rational functions (functions, again!) on this set and then prove that this set is actually isomorphic to an "actual" curve.
b) Lüroth's theorem, which asserts that if $k$ is an algebraically closed field and $k\subsetneq K \subseteq k(x)$ is a subfield of the purely transcendental extension of $k$ of transcendence degree $1$ is isomorphic to $k(x)$ (i.e. $K\simeq k(x)$, but not necessarily equal). You might agree that this is a "purely" algebraic question. Yet, one way to prove this is to observe that $K$ corresponds to a smooth projective curve, say $X$, over $k$ (for instance by a)!) and the inclusion $K \subseteq k(x)$ induces a morphism $\mathbb P^1\to X$. The latter implies that $X\simeq \mathbb P^1$ and hence the statement of the theorem.
These two examples also provide examples to your extended question:
A few things where birational geometry is relevant/interesting/worthwhile:
Resolution of singularities: Given an arbitrary variety $X$, find a proper/projective morphism $\tilde X\to X$ such that $\tilde X$ is non-singular.
This has lots of variants, and the relevant existence theorems are probably the most often used ones in algebraic geometry.
Rationality questions: Which algebraic varieties are birational to projective space? If you want, this is a generalization of Lüroth's theorem. This is also one that one can consider either a purely geometric or a purely algebraic question. Again, lots of variants. Interesting keywords to look up: unirational, rationally connected varieties.
The minimal model program is a quintessential part of birational geometry. It is sometimes considered part of classification, although it is actually really a precursor of that. The main guiding question of the mmp is: How can we choose a nice representative from every birational class?
These are the first three that come to my mind, but others might argue that I am forgetting some. So, let me add some random examples without claiming that they are the most important ones.
Birational geometry comes up in string theory. Physicists like to work with smooth objects, so they are in trouble when they encounter a singular object, which is essentially inevitable. So they want to make is smooth. You can do that either by resolution of singularities (as above) or (sometimes) by deforming it to something smooth. Doing both leads to something they call the conifold transition (see here)
Another place where birational geometry is extremely important is moduli theory. This is another huge subject, so I'm not going to get into it.
In fact, I kind of feel that I could just keep going, so instead I will stop here. Let me just say this: birational geometry is everywhere in algebraic geometry and even beyond that.
To respond to the question in the comments:
I would somewhat disagree with your assessment of the message of the answer on math.se. IMHO, isomorphism and birational equivalence are not competing.
A different way to say what the math.se answer is saying is this: A priori the classification problem is to classify all varieties up to isomorphism. This is essentially impossible in the sense of usual classifications (such as classification of finite simple groups).
The method employed is the following: first find
a nice representative in each birational class, and a way to obtain it, then classify these nice representatives (up to isomorphism).
This does give you a sort of classification of all varieties up to isomorphism. Start with your favorite one, obtain the nice representative, look it up in the classification. Now, you can obtain the original one by reversing the way you obtained the nice representative starting with the one you found on the list.
In fact, this gives you a better classification than if you just had a list, because using this method you might be able to compare two varieties and decide whether they are isomorphic and if not then whether they are birational. But this is not the main point.
The main point is what I said already, isomorphism and birational equivalence are not in competition. They play for the same team.
(Also, just to link this to the above. That "nice representative" I mention here is the minimal model and the way to get it is provided by the mmp.)