The fundamental is the basic "lowest" frequency contained when an instrument is playing a note. E.g. concert A = 440Hz.
The overtones are all frequencies produced which are higher than the fundamental; these are a big part of determining an instrument's sound.
The harmonics are a subset of the overtones; harmonics are those overtones which are "harmonically related" to the fundamental. That is, harmonics are integer multiples of the fundamental. So if fundamental = F Hz, then 1st harmonic = fundamental = 1*F, second harmonic = 2*F, third harmonic = 3*F, etc. The overtones of most, but not all, instruments tend to be harmonic, so the terms are often confused.
An octave is an interval of frequencies of the form [F, 2F]; i.e. the highest frequency is twice the lowest. So the frequency n octaves above F is 2^n * F, the frequency n octaves below F is 2^-n * F.
Even if an instrument's lowest possible fundamental is, say, 40 Hz, it may still produce frequencies near zero Hz when a string/hammer/whatever is struck. These are sometimes called attack transients; the math tells you that an impulse sound (like a hand clap, drumstick strike, etc) will in fact contain all frequencies, including those extending down to zero Hz. Whether or not these appear on any given recording depends on a lot of things, such as how closely the instrument was miked, use of EQ/filtering, etc. However, if you mike an instrument closely enough, you will see that you definitely capture transients which are well below the fundamental.