This means that Bézout's identity holds for every pair of elements, and that every finitely generated ideal is principal.
The expression of the greatest common divisor of two elements of a PID as a linear combination is often called Bézout's identity, whence the terminology.
In particular, in a Bézout domain, irreducibles are prime (but as the algebraic integer example shows, they need not exist).
Finally, if R is not Noetherian, then there exists an infinite ascending chain of finitely generated ideals, so in a Bézout domain an infinite ascending chain of principal ideals.
Let R be a Bézout domain and M finitely generated module over R. Then M is flat if and only if it is torsion-free.