The composition of the eraser does cause some differences in how the graphite is removed, but in general I've found that a dabbing twisting motion is best regardless of what the eraser is made of.

I find soft putty erasers to be the best for soft pencils, and a hard grit eraser best for H4 and harder, but be gentle as it can shred the paper fibers.

The Staedtler Mars Plastic is a good general purpose eraser, in my opinion.

And as a final tip, keep the eraser clean: I find rubbing the built up graphite off with your thumb to be a good action prior to using it.