I am slightly confused about what the name of the bond is between the phosphate and sugar within a nucleotide. All my research comes up with is a phosphodiester bond being the backbone of DNA. But within a single nucleotide, would we perhaps call it a phosphoester bond (all of my searches of phosphoester bond correct it to phosphodiester, so I don't think the term 'phosphoester' is in use...); or maybe it would be an O-glycosidic bond as it is a sugar molecule covalently bonded to another molecule via an O atom?
Compounds formally obtained by esterification of the 3 or 5 hydroxy group of nucleosides with phosphoric acid. They are the monomers of nucleic acids and are formed from them by hydrolytic cleavage."
Phosphoester or phosphoric ester means an ester of phosphoric acid.
Expanding on WIYSIWG's correct answer:
Each nucleotide contains one phosphoester bond (between a phosphate O and sugar 5'-C). Additionally, two nucleotides are connected by one phosphoester bond (between a phosphate O and sugar 3'-C). So in a polymer of multiple nucleotides (DNA, RNA), the repeating monomer unit contains two phosphoester bonds, on "top" (5') and "below" (3') the sugar. Since the repeating unit contains two phosphoester bonds, and the phosphates alternate with the sugars in the sequence, we call this a sugar-phosphate backbone held together by phosphodiester bonds/linkages.
Schematic of single nucelotide featuring vertical line as phosphoester bond (to 5' C):
$P \\ | \\ [sugar]-[base]\\ \\$
Two nucleotides, each with an internal phosphoester bond and with a new phosphoester bond connecting them (spanning 5' of one sugar to 3' of the next):
$P \\ | \\ [sugar]-[base]\\ |\\ P\\ | \\ [sugar]-[base]\\ \\$
Repeating unit in nucleotide polymer, with alternating sugar-phosphate backbone and one phosphodiester bond/linkage per unit/nucleotide residue (the ... ellipses indicate pattern continues for the entire nucleic acid polymer, with hydroxyls capping the very ends):
$...\\ P\\ | \\ [sugar]-[base]\\ |\\ ...\\$