You’re bound to come across these two verbs if you look up to break in Russian.
The verb разбить [imp: бить] means to break, in the sense of to break into pieces, to shatter. This verb, as a result, is key for cups, glasses, dishes and really anything that shatters.
Чашки выпали у неё из рук и разбились.
The cups fell out of her hands and broke [shattered].
Я был так сердит, что разбил все тарелки.
I was so mad that I broke [shattered] all the plates.
A romantic phrase utilizing this verb:
Она разбила мне сердце.
She broke [shattered] my heart.
And a more advanced usage:
Разбитая любовь никогда не клеится.
A broken [shattered] love never mends itself.
The verb сломать [imp: ломать] means to break, in the sense of something that doesn’t shatter, a stiff object that breaks in two, less severe. It can also mean to break down.
Григорий отсутствовал на работе, потому что у него машина сломалась.
Grigorii missed work because his car broke down.
Кирилл сломал ногу, играя в хоккей.
Kirill broke his leg playing hockey.
And the useful expression ломать голову над + instr. means to rack one’s brains over something.
Я сломал голову над этим сложным заданием.
I racked my brains over this complex task.