The goal of the mmp is to find a representative in every birational class that for some reason may be considered nice.
For curves the answer is clear, there is a unique smooth projective representative and by any consideration that is the one that represents the class best.
For surfaces this gets complicated as there are non-trivial birational maps between smooth projective surfaces. However, since they is always a combination of blow-ups and blow-downs it is relatively easy to keep order.
Observe that a $(-1)$-curve is usually defined as a curve isomorphic to $\mathbb P^1$ having self-intersection number $(-1)$. A perhaps better definition that points to higher dimensional equivalents is that a $(-1)$-curve is a curve isomorphic to $\mathbb P^1$ having an intersection number $(-1)$ against $K_X$ where $X$ is the surface on which the curve lives. These two definitions are equivalent by the adjunction formula, but the latter one has the advantage that it does not depend on $X$ being a surface.
Let's take a look at a minimal model of a surface. Why do we pick that as our representative? In some sense there might be other ways to pick a representative, but one might argue that a minimal model is the "simplest" model that is still smooth (make a note of this, we will realize later that here smoothness is actually something else in disguise). Castelnuovo's theorem about blowing down $(-1)$-curves says that we can "get rid of them", so why not do that. Let's contract everything we can. It can be proven relatively easily that contracting a curve that is not a $(-1)$-curve will lead to singular points.
OK, so the strategy is to contract as much stuff as we can and hope that this way we get a reasonable theory. The second definition of a $(-1)$-curve suggests that to find what we can contract is through $K_X$, that is, things that can be contracted and not cause too much trouble are $K_X$-negative. In fact there is a more precise way to say this, but let me not get into technical details now.
So, either this way or already for surfaces one realizes that what makes a minimal model tick is that $K_X$ is nef, that is, intersecting with any proper curve gives a non-negative number. So, now you say that $\mathbb P^2$ is a minimal surface but $K_X$ is negative ample so this is pretty far from being nef. Yes, in the modern terminology of the mmp, $\mathbb P^2$ is actually not minimal. The claim is that every variety is birational to one that is a series of Fano fiber spaces over a minimal variety.
Perhaps I should mention an interesting example here, I think it is due to Iitaka, or someone from his school: Take a $3$-dimensional abelian variety $A$ and mod out by the involution $(-1)\cdot$. Resolve the resulting $64$ double points and call the result $X$. Then it is relatively easy to prove that $X$ is not birational to a smooth projective variety with a nef canonical bundle. At the time this was thought of as proof that minimal models did not exist in higher dimensions, but then Reid and Mori realized that it only means that minimal models need not be smooth. (N.B.: The above accepted answer of David starts by saying that a minimal model should be non-singular. He says it is too ambitious, but it may not be absolutely clear to everyone that this means impossible--as stated. And I promised a comment about why $2$-dimensional minimal models are smooth. The thing is, minimal models have no worse than terminal singularities. It turns out that terminal singularities are smooth in codimension $2$, so in particular a $2$-dimensional terminal singularity is actually smooth. So, one could argue that even minimal models of surfaces have terminal singularities, that is, that's the natural class of singularities for a minimal model. It just so happens that in dimension $2$, these singularities are indistinguishable from smooth points.)
Anyway, so we want $K_X$ to be nef and to obtain this we want to contract curves that are $K_X$-negative. It so happens that this can be done, but this is the result of some very deep results by Mori, Kollár, Kawamata, Reid, Shokurov and others. Now, already in dimension $2$ we get more than just blowing down $(-1)$-curves: the ruling map of a ruled surface and $\mathbb P^2$ mapping to a point are both contractions of $K_X$-negative curves. In general this is how we might end up with a Fano fibre space. It is possible that the contraction of a $K_X$-negative curve is not birational, but that's OK. This really means that the cycle class of that curve covers the entire $X$ and in particular it is uniruled and will never have a minimal model in the sense of $K_X$ being nef.
If the contraction is birational, then there are still two possibilities: it is a divisorial contraction or a small contraction. The former means that the exceptional set is a divisor, the latter that it is smaller than that. Now, already the former can bring in singularities, but they are not so bad and the program can continue.
When the contraction is small, there are several problems. Simply put the singularities become too bad. The badness mainly manifests itself in the singularity being non $\mathbb Q$-Gorenstein, that is, $K$ will no longer be $\mathbb Q$-Cartier which is otherwise needed. And it's not that this may be so, but it will be so for certain: if the target had a $\mathbb Q$-Cartier $K$, it could be pulled back, at least numerically (or some power could be pulled back). The pull-back would have to agree with $K$ upstairs since the map is an isomorphism in codimension $1$. However, a pull-back is necessarily trivial on the fiber of the map, but the fiber was chosen to be $K$-negative. This is a contradiction, so the target cannot have a $\mathbb Q$-Cartier canonical sheaf.
Flips were invented to remedy this situation: the original reason for wanting to contract was to "get rid" of this $K$-negative curve, so let's get rid of it a different way. Being $K$-negative is really a curvature condition and it says something about the normal bundle of the curve inside the variety. (OK, you have to adjust this slightly for singularities, but I am not writing a precise paper here). So, the idea of the flip is this: let's change the normal bundle of the curve. So, let's "cut it out" and put it back with the opposite normal bundle, so in a "flipped" way. (Remark: this is the $3$-dimensional picture, in higher dimensions it's not just curves that get flipped, but this may be better delegated to another place).
I guess I wrote a whole bunch of things just to say that and some people have said similar things already, but perhaps this little essay gives some new insight.
To answer your question about whether a similar construction exists elsewhere, the answer is "yes". A "flip" is like a "surgery" in topology. But I am no expert on that. Actually, just to include a disclaimer: I am not claiming to be an expert on flips either.