Syncopation, and the accent that lands where the pulse is not
The accent displaced off the pulse, which only exists against an established metre and is the rhythmic mechanism of nearly all Brazilian popular music.
Count "one, two, one, two" tapping your foot. Now sing a note that comes in between the one and the two, and stays there, holding through the two without attacking again. You have just made a syncopation: an accent where the pulse is not. That is all it is, and it is the basis of nearly all Brazilian popular music.
The pulse first
Syncopation only exists against an expectation. With no established pulse there is no "wrong" place for the accent to land, and what you hear is just some rhythm. So it needs two layers: one holding the pulse — the foot, the surdo, the bass — and another contradicting it.
That is the pulse: nothing displaced, one attack per beat. Everything below is measured against it.
The displacement
Divide each beat in two and attack on the second half, holding through the next beat:
Quaver, crotchet, quaver. The crotchet begins on the "and" of one and ends on the "and" of two; beat two passes inside it, with no attack. It is the simplest syncopation there is, and it is the cell that turns up everywhere, from samba to funk.
Written with a tie, so that beat two is visible where it belongs:
Same sound, different spelling. The second is easier to read in busy bars, because every beat stays visible; the first is shorter. Both are correct, and both are syncopation.
Across the bar line
The displacement can push a note into the next bar. It begins on the last "and" of one bar and lasts into the first beat of the next, and the strong beat passes with no attack:
Here the tie is compulsory, because no note value can cross the bar line. And the effect is the strongest of all: beat one, the most predictable in the bar, arrives with no attack. The ear feels the pulse and feels the note contradicting it at the same time.
With a rest
Syncopation is also born from a rest on the strong beat. The attack falls after it, on the offbeat, and it is the beat's only attack:
Rest, quaver, rest, quaver: nothing but offbeats. It is the right-hand groove in a great deal of guitar accompaniment, and what makes it syncopation is the rest that leaves the pulse without an attack.
The Brazilian cell
Put the displacement together with a division into semiquavers and out comes the cell that organises samba, choro and maxixe: semiquaver, quaver, semiquaver, accenting the middle of the beat.
The first beat has three attacks, and the middle one is the longest; the second beat has two equal ones. Play that over a pulse of two and the groove appears by itself. No theory is needed to recognise it: it is the difference between a samba and a march played at the same tempo.
What to take from here
Syncopation is an accent off the pulse, and so it needs a pulse to exist. It is written with a note that starts on the offbeat and holds through the beat, with a tie when it crosses the bar line, and with a rest when the strong beat falls silent. The semiquaver–quaver–semiquaver cell is its Brazilian form, and tempo has nothing to do with any of it: the groove is where the accents land, not how fast the pulse goes by.