The process is known as vitrication.
In commercial nuclear fuel, the uranium bearing fuel is in the form of ceraminc UO2, a metal oxide. Most of the fission products are entrained as metal oxides, although metal inclusions are found, and there are other elements like I, Br (halides) and Xe, Kr (noble gases) present. The fuel and fission products are encased/enclosed (or otherwise hermetically sealed) in a tube (cladding) of Zirconium alloy (usually, but it could also be a stainless steel) which is seal-welded at both ends (hopefully) with barstock of the same alloy comprising the cladding.
In vitrification, the fuel is oxidized, and then mixed with glass materials so as to form a type of glass which entrains any fuel (if it is not recovered, as in recycle or reprocess) and fission products. The vitrified was is then encapsulated in a corrosion resistant material (e.g. stainless steel or nickel-based alloy), which can then be placed in a repository.
A related process is Synroc, in which a composition representing thermodynamically stable synthetic igneous rock is used instead of glass. The theory is that Synroc is less likely to allow fission products and transactinides to leach into the envirnoment if there is a subsequent breach of the repository and waste containers.