I think Greg is much closer to your solution than I am.
A phase is, and I am no metallurgist, in this context a lattice configuration.
BCC-Body Center Cube, FCC-Face Center Cube, HCP Hexagonal Close Packed, etc.
Many alloys will, solidify from the liquid as one 'phase' and then change its lattice configuration upon cooling. Carbon Steels for example. Carbon steels go BCC, FCC, BCC.
Many alloys will not. Ni alloys do not undergo phase transformation. They solidify FCC and stay that way.
Grain size is based upon a variety of things. Chemistry and cooling rate being predominant. If a material of a large grain size is reheated through a phase transformation temperature and then cooled faster than its original cooling rate, then you can realize grain refinement.
Embrittlement (and you can probably google a definition more sophisticated and accurate) is a term used to identify a deleterious change in the ductility of a material, and manifest in either bend or tensile testing or other mechanical testing methods.