We have a pretty good understanding of every operationally meaningful aspect of measurement.
The key difference between quantum mechanics and other wave theories is that the wave interference happens in phase space, rather than physical space. In other words, quantum interference happens between different quasi-classical histories of the entire world that end up with the entire world in the same state. The difference between the histories is called welcher Weg, or which-path, information. If you see interference, there must be no record anywhere in the world of which path the world "really" took (the welcher Weg information was lost). If you arrange to record that information in any way at all, you won't see an interference pattern.
This means that in the traditional double-slit experiment, it's not necessary for a person to "observe" a photon going through one of the slits, or even for anything you'd conventionally call a measurement device to be there, in order to make the interference pattern disappear. An arbitrarily tiny change caused by the passage of the photon will do, as long as it's unambiguous and persists after the photon is gone. In quantum computing terms, copying a qubit with a CNOT gate, and then never using the copy, is equivalent to measuring the qubit in the computational basis.
However, such tiny changes are not normally called measurements, and the reason is that in principle you can undo them and get the interference pattern back. Measurements are supposed to be final: wavefunction collapse can't be undone. This leads to the definition of a measurement as a thermodynamically irreversible version of the above. A photon striking a silver halide crystal (and darkening the whole crystal) is a measurement. Any but the simplest photon detector at a slit will measure the photon as well, even if it has no visible readout, since even changes like a slight resistive heating of a wire, leading it to emit more blackbody radiation, will unerasably record the photon's passage.
The work on environmental decoherence only goes back a few decades, but I think this notion of measurement was understood much earlier. Bohr referred to "irreversible amplification" at least in the 1950s (quoted in physics/0002049).
This doesn't answer the question of how one measurement outcome becomes real and the rest don't, but if that process, whatever it is, happens after the irreversible amplification then it is inaccessible to experiment, because we would have to reverse the irreversible process to look for interference effects. So we may plausibly already know as much about measurement as we will ever know.
As for "observer", that's a meaningless term unless it's merely defined to mean "someone who makes measurements". Only measurement has a clear meaning.
