。
Mother will probably make me a birthday cake.
Literal
Mother [subject-が] me [to-に] birthday cake [obj-を] will-make-for-me probably.
~だろう is the plain-form conjectural ('probably,' 'I suppose'), the casual counterpart to でしょう. Attached directly to the plain form of a verb (作ってくれる + だろう). The whole sentence is anticipation rather than fact. バースデーケーキ is the loanword — Japanese has both 誕生日ケーキ (Sino-Japanese + loan) and バースデーケーキ (fully loaned), roughly interchangeable.