My mother gives my younger sister 2,000 yen a month.

Literal

Mother [topic-は] younger-sister [to-に] month [per-に] two-thousand-yen hand-over-[ongoing-ている]

月に二千円 = '2,000 yen per month' — に marks the interval. 渡している = 'is giving/handing over' (ongoing habit). 妹 = younger sister.