I was watching this video about Just-intonation tuning system. In this video he is explaining how the harmonics work. He is explaining that the first harmonic is created by dividing a string into half, which has the lowest frequency. Then by dividing more, we will get higher frequencies. He is also explaining that in the major 2nd the string is divided into 16 parts and has the 15th harmonic, and major 3rd is divided into 9 divisions and has the 8th harmonic. I understand that in a guitar, the closer the finger to the neck of the guitar, the lower the frequency would be. But I don't understand why major 2nd has more divisions than major 3rd? Shouldn't a major 3rd be higher in frequency than a major 2nd? Maybe he means that there is a major 2nd an octave higher than this major 3rd?
First, you seem to have associated the ratios with the wrong intervals: 16:15 is the ratio of a minor second, not a major second. The major second is 9:8 or 10:9, and the major third is 5:4.
Second, you seem to be confused about how the intervals are derived from the harmonics. Consider the 9:8 major second. This is the ratio between the 9th harmonic and the 8th harmonic. Now consider the 16:15 minor second, which is the ratio between the 16th and 15th harmonics. Yes, the 16th harmonic is an octave higher than the 8th harmonic, but the 15th harmonic is also nearly an octave higher than the 8th harmonic, and the distance between the 16th and 15th harmonics is rather smaller than the distance between the 9th and 8th harmonics. Therefore, a minor second is smaller than a major second.
The fundamental of the overtone series that the ratio implies is not particularly important. For example, suppose you want to calculate a 16:15 semitone above A=220. Hz. To do that, you multiply 220 Hz by 16 and divide by 15, yielding 234.67 Hz, which is roughly the same as the first-fret B♭ on a guitar. To calculate a 9:8 whole tone above A=220 Hz, multiply by 9 and divide by 8, which gives 247.5 Hz, a higher pitch, roughly the same as a second-fret B.