The oboe typically plays Fs in three octaves. The lower two have a variety of available fingerings, which can be a challenge for new oboists to navigate, particularly because the available fingerings change depending upon the make of the instrument.
A typical “budget” student model instrument, for example, uses the following fingerings. (For all fingerings given in this article, the one shown corresponds to the lower octave; the higher octave is achieved by adding the first [thumb] octave key.)
The “right” F is the basic choice, to be used in almost all cases where it is possible to do so, as the tone produced by this fingering tends to be the best match to the tone of the surrounding notes.
The “forked” F tends toward a sound that might be described as “muted” or sometimes even “fuzzy,” and should therefore generally be avoided where possible (unless the muted or fuzzy sound is desirable for the musical situation—I do like to use the forked F, for example, in the beginning and ending sections of the second movement of the Saint-Saëns sonata).