I'll often hear tunes in which the first note is upbeat and the second is downbeat. When is it sycopation and when is it not? For one of many examples, take "Amazing Grace." The "A-" is a weak beat and "-mazing" the strong beat. Yet I hear others claim that this is not syncopation. Or take "Somewhere Over the Rainbow" in which the second note jumps a full octive. "Some-", which is the first note, is weak, and "-where" is a downbeat. Isn't that syncopated? Others claim that I don't understand the principle but yet are not able to explain what I don't understand. I sure need help on this one. Incidentally, I play the clarinet and sax at the intermediate level, and beginner's piano.
Let's forget about bar lines and look at some stuff where we're just marking stressed notes with accent marks. In everything that follows, the quarter note has the beat.
"Amazing Grace" goes like this:
Notice that a stressed note comes every third beat; the only exceptions are when there's a long note at the end of a phrase, and even then they're separated by a multiple of three beats. So if we put the bar lines back, in the standard places (i.e., in 3/4 time with a 1-beat pickup to reflect the fact that the first quarter note is unstressed), they're describing exactly what's going on with the stress, and therefore there's no syncopation going on.
In contrast, "Somewhere Over the Rainbow" might be sung like this:
This pattern of stresses is almost as regular as the previous one, but not quite. The first three notes, "some-", "where", and "o-", are all stressed, even though they occur at irregular intervals. After that, though, things settle down; there's a stressed note every two beats. We could insert bar lines to write this in 2/4 or 4/4 and capture the locations of all these stresses, except that the syllable "where" is stressed despite the fact that it occurs on an ordinarily unstressed beat. That is, the "where" breaks the ordinary pattern of stresses, which makes it a syncopated note.
The basic understanding of syncopation is when a note starts off-beat, and hanging over to the next off-beat. It is most common to say it is syncopated when it is a pattern that breaks with the basic beat, and not so much when it is just emphasize on the weak beat.
It is often used as effect to create a rhythmic tension that leads forward to and resolves in a non-syncopated pattern.
Amazing Grace is definitely not syncopated. Syncopation generally refers to rhythms emphasizing the beats that are expected to be the weak beats (for example 2 & 4 in 4/4). Or emphasizing notes off the beat entirely (for example emphasizing the + of each beat in 4/4 such as one AND two AND three AND four AND).
There are many forms and types of syncopation, but basically it involves upsetting the expected strong beat pattern.
Your example of the "where" in Somewhere over the Rainbow does exemplify syncopation, at least as I'm hearing it in my head "where" lands on and emphasizes beat 2 in 4/4. (Perhaps some singers/arrangers place "some" on beat one and "where" on beat three, which would not be a syncopated rhythm.)