Let's go right to some examples. Rock and roll is partly based on a very simple kind of syncopation. In a measure of four beats the normal ones to stress are one and three: ONE two THREE four. In rock and roll, instead the two and four beats are stressed. This is known as the "back-beat". Here is an example:
It would be a creative thing to try and do a rock song that didn't do this and in fact Cream did. Here is "Sunshine of Your Love":
What gives it its primal effect, apart from the 'lick', is the strong beats on one and three, possibly inspired by Ginger Baker's interest in African drumming. Neophyte rock drummers are sometimes very puzzled to play this one because they are so thoroughly programmed into putting the stress on the backbeat.
This is syncopation on the basic level of style or genre. Another example would be the Baroque sarabande, a slow dance in 3/4 time. Normally in triple time the first beat is stressed, the second beat is neutral and the third beat lifts--the 'upbeat' (meaning that in a dance, the dancer would typically lift their foot on this beat so as to put it down on the first beat). But in the sarabande, it is the second beat that is stressed: one TWO three. An example:
The common metric technique based on syncopation is called 'hemiola' and involves turning two measures of 3/4 into one measure of 3/2 by using syncopation: ONE two THREE one TWO three. This was often used at final cadences in Baroque music.
But syncopation is found on all rhythmic levels, even within beats. So it is not only a metric technique, but also a rhythmic one on the level of melody and a harmonic one as well, as intense harmonies are often placed on weak beats. In the following example, the third movement of a Beethoven quartet, the first variation uses some intense accents on the weak parts of the beat at the end of the first section. The effect is around 1:53 in this clip:
The only kind of context in which you cannot use syncopation is one where there is no context of strong and weak beats and hence no expectation to play with. This is the problem with the jagged rhythmic textures of high modernism because they are so fragmented that the listener has no sense of a regular beat. Ironically, these pieces that look extremely 'rhythmic' on paper, really have little rhythmic effect because they sound random.