Multiple transition phenomena in divalent Eu compound EuAl$_4$ with the tetragonal structure were investigated via the single-crystal time-of-flight neutron Laue technique. At 30.0 K below a charge-density-wave (CDW) transition temperature of $T_{rm CDW}$ = 140 K, superlattice peaks emerge near nuclear Bragg peaks described by an ordering vector $q_{rm CDW}$=(0 0 ${delta}_c$) with ${delta}_c{sim}$0.19. In contrast, magnetic peaks appear at $q_2 = ({delta}_2 {delta}_2 0)$ with ${delta}_2$ = 0.085 in a magnetic-ordered phase at 13.5 K below $T_{rm N1}$ = 15.4 K. By further cooling to below $T_{rm N3}$ = 12.2 K, the magnetic ordering vector changes into $q_1 = ({delta}_1 0 0)$ with ${delta}_1$ = 0.17 at 11.5 K and slightly shifts to ${delta}_1$ = 0.194 at 4.3 K. No distinct change in the magnetic Bragg peak was detected at $T_{rm N2}$=13.2 K and $T_{rm N4}$=10.0 K. The structural modulation below $T_{rm CDW}$ with $q_{rm CDW}$ is characterized by the absence of the superlattice peak in the (0 0 $l$) axis. As a similar CDW transition was observed in SrAl$_4$, the structural modulation with $q_{rm CDW}$ could be mainly ascribed to the displacement of Al ions within the tetragonal $ab$-plane. Complex magnetic transitions are in stark contrast to a simple collinear magnetic structure in isovalent EuGa$_4$. This could stem from different electronic structures with the CDW transition between two compounds.
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