The harmonics are defined as the sequence of frequencies obtained by multiplying the fundamental frequency by whole numbers. So if the fundamental frequency is f, the harmonics are given by fn = nf and the first harmonic (n=1) coincides with the fundamental. The overtones on the other hand are defined as the sequence of frequencies above the fundamental - literally overtone means above the tone and the tonic is just another name for the fundamental (in that context).
For a string instrument, the overtones coincide with the harmonic series so we have, as AlephZero explained the identification.
Fundamental frequency = first harmonic
First overtone = second harmonic
Second overtone = third harmonic
etc.
That's also the case for open wind instruments such as a flute. But closed wind instruments (blowing off the top of a beer bottle for instance) only produce odd harmonics so we have a different identification table
Fundamental frequency = first harmonic
First overtone = third harmonic
Second overtone = fith harmonic
etc.
The overtones produced by percussion instruments are in general much more complex and in general do not coincide with the harmonic series at all.
That's why drums are - well - less harmonic in nature and are not used to create harmony (musical cords). For instance in a piano (string instrument) if you play a tonic, a major third and a fifth at the same time, you create a nice sounding chord called the major cord. That works because the 2nd overtone of the tonic coincides with the 1st overtone of the fifth and the 4th overtone of the tonic coincides with the 3rd overtone of the major third, and the 4th overtone of the fifth coincides with the 5th overtone of the major third. All these matching overtones bind the different notes into a harmonic chord. Since a drum's overtones are not harmonic, trying to play chords with drums doesn't really work. The overtones don't match and the chord wanna-be just sounds like a muddled mess. Drums are for rhythm - not harmony.