Two generalized homology theories$E_\bullet$, $F_\bullet$, hence spectra$E$, $F$ are called Bousfield equivalent if the homology groups of both always vanish simultaneously, hence if for every homotopy type/spectrum$X$ we have $E_\bullet(X) \simeq 0$ precisely if $F_\bullet(X) \simeq 0$.

Examples

Global arithmetic fracture square

There is a Bousfield equivalence

$S (\mathbb{Q}/\mathbb{Z}) \simeq_{Bousf} \vee_p S \mathbb{F}_p$

Aldridge Bousfield, The localization of spectra with respect to homology , Topology vol 18 (1979) (pdf)

and was named such in

Douglas Ravenel, Localization with respect to certain periodic homology theories, American Journal of Mathematics, Vol. 106, No. 2, (Apr., 1984), pp. 351-414 (pdf)