Assuming that the container is closed from all sides, including the top, what happens also depends on if it is full, or if there is some space filled with air, some other gas, or vacuum.
If there is space, the container will move. It will also change the center of mass as the liquid splashes around.
If there is no space and the container is full of liquid, other answers describe what happens.
This is important when transporting liquids on boats. If the liquid can move around, the motion can flip the boat. Containers get designed to minimize the chance of that happening.
There is no space, unless you left a vacuum, and sealed the container under this vacuum. And even then, the liquid will evaporate to fill the empty space with gas, up to some pressure (that depends on the composition of the liquid and temperature).
That said, if there are thus more than one phase in this container, what happens is dictated by Archimedes. When one phase is less dense than the other, then it will be sujected to a force equivalent to the "weight" or acceleration of its volume filled by the more dense phase. That, plus the surface tension of the interphase surface will dictate how bubbly the phases will change position during your shaking acceleration. This can lead to the creation of lots of small bubbles, up to mixing an "emulsion" with the right surface tension.
If there is space, the container will move. It will also change the center of mass as the liquid splashes around.
If there is no space and the container is full of liquid, other answers describe what happens.
This is important when transporting liquids on boats. If the liquid can move around, the motion can flip the boat. Containers get designed to minimize the chance of that happening.