If you play an A on the third string, second fret of your guitar, you will be playing the frequency 440 Hz (approximately, depending on tuning). Because of the physics of your guitar though, this is not the only frequency that the guitar would produce. Parts of the guitar and parts of the string will vibrate at frequencies other than 440 Hz. If those frequencies are higher than 440 Hz they are called overtones. If those frequencies are lower than 440 Hz they are called undertones. More than likely you will get both. The frequency 440 Hz is called the fundamental frequency. Some overtones are special, as they fit nicely with the fundamental frequency. You get those frequencies if you multiply the fundamental frequency by 2, 3, 4, and so on. These are the harmonics of the fundamental frequency.
The harmonics of a frequency are frequencies that are integer multiples of that frequency.
For example, the frequencies 880, 1320, 1760, and so on are harmonics of the fundamental frequency 440 Hz. Harmonics fit nicely with the fundamental frequency. If 440 Hz is A, then 880 Hz is A an octave higher, 1320 Hz is approximately an E (the fifth of the A in the higher octave), 1760 Hz is another A, and so on.
Even order harmonics and odd order harmonics
Even order harmonics of some frequency (the fundamental frequency) are frequencies that are 2, 4, 6, and so on times the fundamental frequency. Odd order harmonics of a frequency are frequencies that are 3, 5, 7, and so on times that frequency.
For example, the harmonics (even and odd) of A at 440 Hz are 880 Hz, 1320 Hz, 1760 Hz, and so on. The even order harmonics of 440 Hz are 880 Hz, 1760 Hz, 2640 Hz, and so on. The odd order harmonics of 440 Hz are 1320 Hz, 2200 Hz, 3080 Hz, etc.