When attempting to explain the overtone series, things can get abstract and a bit complex very quickly, so I will attempt to provide an explanation that is as simple as possible.
Let’s start by saying that whenever you hear a single sound, say one note on a piano or sung by a voice, you are not only hearing one sound but rather a combination of various sounds. Even though in music you would call that one note (or one tone), in reality it is a combination of tones.
So let’s say that the note is a C on a piano keyboard. The note we identify as C is only the fundamental or lower tone of an ascending series of tones with increasing frequencies. These additional frequencies are called overtones.
The overtone series is a physical phenomenon and the “distances” between the frequencies of each overtone are fixed and can’t be changed.
The overtones follow a specific sequence. When we compare this to music, the sequence is such that if we take C as fundamental (as we said before), the first overtone is said to be the same C (in music, this interval is called unison). The second overtone is again a C but one octave higher (double the frequency). The third overtone is a G above the second C (the musical interval of a perfect fifth). The fourth is again a C, one octave higher than the previous one. The fifth overtone will be an E (major third). The overtone series goes on and it is virtually infinite, with octaves repeating in between other intervals that become less and less consonant as the frequency increases. Interval, in music, is the name given to the distance between two notes. Intervals are as many as the possible combinations of different notes.
It is important to keep in mind that the frequencies of the the notes mentioned above, differ from the frequencies of the same notes on a piano (exactly because pianos are usually tuned to equal temperament)
The “problem” started with the invention of fixed tuning instruments, mainly keyboards. In fact, the voice as well as the instruments of the strings section (violin, violas, etc), some brass instruments such as the trumpet and trombones, ancient instruments such as the horn, are all capable of adjusting the pitch by ear, as they play.
Fixed tuning instruments such as keyboards and guitars can’t do that. Once they are tuned, each key produces a pitch that can’t be changed, unless re-tuned (yes, one can “bend” notes on the guitar, but it would be impossible to do that for all notes in a piece of music).
So these instruments have to use a “tempered” system, where some of the intervals have to be altered in order to fit into an octave.