Several recent questions have been about tuning. Often an orchestra will tune to the oboe. One reason for this is that oboes are the only instruments that cannot be tuned. 1. What if the oboe was actually out of tune for some reason? 2. When the oboe is not present, who takes over? 3. For piano concerti, would the piano take precedence? And why would all this happen?
Oboes most certainly can be tuned (perhaps less tuning range than other instruments).
If there's a piano (or even worse, an organ :-) ) involved, typically the oboist will tune to the piano first, then let the orchestra tune to the oboe.
String orchestras typically tune to the first violin; chamber ensembles work it out one way or the other. FWIW wind ensembles (aka wind orchestra, concert band) tune to the first clarinet instead of the oboe. Damn if I know why. (Cue the first song from Fiddler on the Roof)
As to why the oboe... well, who wants to anger those who play "the ill wind that nobody blows good" ?
Carl's answer is right: you can tune an oboe and if a fixed tuning instrument (like a piano, an organ, a piano accordion, etc.) is part of the ensemble then the musicians will tune to that.
I had always understood that the oboe was used because of the harmonic composition of its timbre, i.e. that it has a relatively pure pitch and is thus easier to tune to. However Wikipedia's page on the oboe suggests that it is the oboe's secure pitch and penetrating sound, resulting from its conical bore, that make it ideal for tuning purposes.
This is a slightly different take on the situation.
Every musician today owns, or should own, a tuning device. The instrument should already be in tune when you get up on stage, we tune backstage. So the initial tuning the audience hears is only for getting confidence that nothing is wrong. The oboe player, unless he has perfect pitch, uses a tuning device to play spot on. We very seldom use a tuning device during playing, instead we listen to each other. I tend to use it when playing contra bassoon, as the low tones can be extremely difficult to hear (beeing as unobrusive as I can).
During the playing we musicians adjust the toning when necessary. Often this means adjusting even single notes to the harmonic function of the note or to other players. As exemple, a harmonic major third is intoned quite differently from a harmonic minor third. (I tend to illustrate this on a piano. Press a C and next higher G. Now sing the major third until it sounds good. Compare with pressing the E. Most people sing a much lower tone than the E on the piano. Same with minor Eb which people sing much higher).
Of course, when playing with fixed pitch instruments lika a piano or the organ we simply adjust to that. It can be interesting to play with organs as they often enough are not tuned to modern standard pitch. (It can be expensive to retune a few thousand pipes). In churches before 1930 or so, the organ tunings varied considerably.
During the concert tuning can be done again. This especially if the room gets warmer during the concert. Wind instrument tends to go up, strings tend to go down when the room gets warmer. During, say, a symphony the players will need to compensate individually. Between symphonys it helps to tune again.
The oboe player checks against a tuning fork (or perhaps, these days, an electronic device). The rest of the orchestra check against the oboe - but it IS only a check, they have already tuned by their own, various, methods. And if there's a piano, it will have been tuned to the agreed pitch.
Why oboe? It's a good, clear sound. But the oboe isn't the primary reference, his tuning fork is. No woodwind instrument likes to be 'thrown out' by having to play at a different pitch than it was built for - even if the reed can be 'pulled out' a bit, the tone holes stay the same distance apart. So the woodwinds possibly benefit more than other instruments from standard pitch.