Frequency is simply the number of oscillations that a wave goes through per second. Any wave which is periodic has a frequency. But usually in music, use of the term is limited to talking about sine waves, so if you hear something about a wave of frequency x, it usually means a sine wave with that many oscillations per second.
Any arbitrary wave, whether periodic or not, can be constructed by adding up sine waves of various frequencies in varying amounts (that is with various amplitudes). What the Fourier transform does is tell you which frequencies to use, and with which amplitudes, to create any given wave. The fast Fourier transform (FFT) is a particular algorithm that computes the Fourier transform of a wave, given the data representing the amplitude of the wave as a function of time.
When you hear a musical note played by an instrument, it doesn't consist of just a single frequency. Instead, what you get is a combination of different multiples of a fundamental frequency, in different amounts. For example, a flute playing a particular note might produce a combination of
- 440 Hz with amplitude 1
- 1320 Hz with amplitude 1/2
- 2200 Hz with amplitude 1/3
and so on. On the other hand, a trumpet playing the same note might produce a combination of
- 440 Hz with amplitude 1
- 880 Hz with amplitude 1/2
- 1320 Hz with amplitude 1/4
- 1760 Hz with amplitude 1/8
and so on. (Those are not the actual relative amplitudes for those instruments; I just made up some example numbers) So in your tuner application, when you run the FFT on incoming data, you will find multiple peaks in the output at various frequencies, depending on which instrument is being tuned. The point is that the output of the FFT will not just be a number; it won't just tell you "this instrument is playing a note at 440 Hz."
Now we get to pitch, which is a slightly more nebulous concept. The pitch of a note is basically what a person actually hears when exposed to that note. For many instruments, the pitch is correlated to the fundamental frequency being emitted by the instrument. But depending on the relative amplitudes of the higher frequencies, a person might perceive two instruments to have different pitches even if they are actually playing the same note.
Fortunately, if you're just making a simple tuner, you don't have to worry about pitch at all. The point of a tuner is to minimize beats between different instruments, and beats are caused by the actual frequencies, not the perceived pitches. A trumpet and a flute both playing with a fundamental frequency of 440 Hz will not exhibit beats because the differences between all their frequencies are multiples of 440 Hz, even if the untrained ear might think one of them is higher-pitched than the other.
