"Can I borrow your silver knife?"
-- Harry to Hermione while dealing with sopophorous beans in Potions class (HBP9)
A short silver knife is specialized equipment for potions work (HBP9). Students typically have these in their school potions kits (HBP26).
Students cut up many ingredients, but don't necessarily use a silver knife. Harry, for example, doesn't have a silver knife with him while brewing the Draught of the Living Death. He cuts up Valerian roots, then has to ask for the silver knife from Hermione to crush the sopophorous beans, which means he was using something different up to that point (perhaps "school knives" laying around in the Potions classroom, rather like using a school broom for Quidditch) and that he didn't have a silver knife himself. Perhaps this is because he wasn't planning on being in the class and hadn't brought his proper tools, since he normally does have that kind of knife:
"I rather think," said Dumbledore, putting his uninjured hand inside his robes and drawing out a short silver knife of the kind Harry used to chop potion ingredients, "that we are required to make payment to pass." (HBP26)
The fact that the knife is silver makes a difference for some potions, judging by the Prince's instructions about crushing the beans for the Draught of the Living Death, and that students sometimes use other implements for their routine work.
Dumbledore carried a short silver potions knife with him and used it to cut his hand in the sea cave. The resulting blood served as an offering to the magical portal which allowed him and Harry to get to the lake and the Horcrux island (HBP26).
Bellatrix used a short silver knife to torture Hermione, then hurled it at Dobby as the elf was using his magic to rescue Harry and the others. The knife killed the elf (DH23, DH24).
