What is meant by phase-change in mantle?
I'd like to add to Gordon's answer. A phase change in this context does not only refer to the change in the state of matter (e.g. liquid to solid) but a change in different solid states as well.

Olivine, $\ce{(Mg,Fe)2SiO4}$, is the stable magnesian silicate in pressures down to those prevailing at a depth of ~410 km below the surface of the Earth. As the mantle is mostly made out of magnesium, silicon and oxygen, this mineral is one of the dominant phases (or minerals) down there. But what happens below 410 km? Using geophysical methods, we know that something changes in the properties of the bulk mantle:



There are "jumps" in the densities and seismic velocities. From experimental work, we know that olivine changes to wadsleyite and ringwoodite at these pressures. Both wadsleyite and ringwoodite are chemically similar to olivine, their composition is also $\ce{(Mg,Fe)2SiO4}$. However, because of the high pressure, their structure changes from the olivine structure to a denser, more compact spinel structure. Note that this is not spinel the mineral ($\ce{MgAl2O4}$), but rather the olivine composition with the ions arranged the same way the ions are arranged in spinel.

At 660 km, the "spinel"-olivine changes to "perovskite"-olivine, causing the second discontinuity. This perovskite is not the mineral perovskite ($\ce{CaTiO3}$), but rather a magnesium silicate with the perovskite structure ($\ce{(Mg,Fe)SiO3}$). Here's a summary of the phases (i.e. minerals) you find in the Earth's mantle:



Both figures are taken from this excellent Uni Münster website, where you can see the high-res figures and read some more about it.