I've been playing the bassoon for just short of two years now and I've been wondering why the standard issue bassoon is called a 'C bassoon' or 'bassoon in C' when the instrument is pitched in F.
There are several misunderstandings going on in the other responses, and in their comments.
First of all, most woodwinds transpose so that they're fingered the most similarly. This is true for flute, oboe, saxophone, the upper register of clarinet (clarinet's first two registers are a 12th apart instead of an octave), as well as standard tin whistle and half of the recorders (soprano, tenor, etc.). With transposition, all of these instruments finger written D, E, G, A, and B the same way, and there are plenty more similarities that aren't quite universal. The target for the transposition is not the lowest possible note for the instrument, and the home key is not the one with the six main fingers down, or the one that's easiest to finger. The pre-Boehm fingering system makes F# easier than F, and usually makes C# easier than C, but these instruments don't call G or D their homes.
If bassoon transposed so that it followed the same fingering convention, it would be an F transposing instrument. That's what Erik is getting at. However, bassoon does not transpose and instead has its fingerings offset from almost every other woodwind instrument.
So why is this? I don't know the history of bassoon as well as I know the history of other instruments, but I can make an educated guess. Bassoon used to come in a few other sizes before we standardized on the modern one. Composers would want to write a single line and let the player figure out how to play it given which size of instrument they had, rather than write separate versions to cover all cases (this is why tuba doesn't transpose, despite Bb, C, Eb, and F instruments being common).
I've never heard of a standard bassoon being called "Bassoon in C" - except perhaps by one particular piece of notation software where an early release had some bugs when deciding if some instruments were transposing instruments or not, and wrongly displaying "in C" for non-transposing instruments.
However there are smaller version of the bassoon, sometimes used by young beginners, called a "quart" or "quint" bassoon or a "tenoroon". These are transposing instruments in F or G, unlike the standard bassoon.
If a piece of music was published with parts for both the bassoon and tenoroon, I suppose the bassoon part might be labeled "in C" to avoid any confusion.
(Incidentally, "tenoroon" is a relatively new name, but the instrument itself is not a modern invention - the original bassoon family was made in as many as 6 different sizes, from soprano to contrabass pitch).
Bassoon is non-transposing and is just called 'Bassoon'. Contra-bassoon is written an octave above sounding pitch (like double bass) and is just called 'Contra-bassoon'.
There are 'Tenoroons', originally in a wide range of sizes, now (Wikipedia tells us) only made in Eb, F and G, and largely aimed as children's training instruments. Maybe these are what you're thinking of?
The music will be for Bassoon in C because in the music a written C corresponds to a concert C (it's concert pitch). The horn (whether an F horn or Bb horn) is written in F because a C in the music corresponds to a concert F.
My take on this. Hobby player on both bassoon and contrabassoon.
I only see the Bassoon called "Basson in C" on websites of music shops not really knowing the instrument. As the Bassoon is only available in one size, there is no need to have the C. We learn to play the Bassoon written as sounding -- this does not in itself imply that the bassoon is a C instrument or in any other key in itself.
I have checked the sources I can find, and I cannot find any reference to bassoons in any other size. Previous instruments, dulcian as example, came in different sizes. The contrabassoon is a different instrument from the bassoon. There are two "modern" versions, that is about mid 19th century and forward. The more common in most of the world is the German version, often called Heckel version after the most known firm. The other version is the french bassoon (I own one of these as well, as a restoration project). Both versions are the same size. From a player perspective, the tone holes and the mechanism differs so you use different fingerings on many but not all notes.
Regardless, what to call instruments usually goes back far into history and is basically not logical. I might make the example of three sizes of trombones: bass, tenor and alto. In old days, sometimes now as well, they were written in three different clefs: bass, tenor and alto. All as non-transposing. A written C sounds as a C. Most tenor trombones are actually Bb instruments. Most alto trombones are Eb (sometimes F). Most bass trombones today are Bb, albeit with larger tubing. In older times bass trombones came in G and F as well (today F trombones are most often called contrabass trombones). Logical -- not!
Among the string instruments, you will not find a C string on a violin, viola or bass <-- note wrong, see comments. There is one on the Cello, but all instruments are non-transposing.
@Scott Wallace-Are you saying that the true "fundamental tone" of a woodwind instrument is defined by the pitch that you get by closing the holes controlled by the index, middle and ring fingers of both the left and right hands? If so, the flute is pitched in D (as was stated by Erik) and the bassoon is pitched in G. Your statement about the bassoon being pitched in F requires the use of "any keys that extend the range downward", meaning the right hand little finger pressing the Low F key.
Further, what if rigged my bassoon's Low F key to close the Low F, E, D and C keys with one press? That does not change my bassoon to a different key; i would still get C if I close only the three fingers of my left hand.
Also, the overall tube length does not determine the pitch of an instrument; flutes used to always have an overall length corresponding to middle C; now Low B feet are common. When I put my Low A bell on my bassoon, I do not change any of the fingerings that I use or incorporate any transpositions. I am not suddenly playing bassoon in A.