Eraser are designed to remove pencil lead (Graphite), ink and even coloured pencil wax material from paper using as light a pressure as needed with little or no smudging, with some being better than others.
There are many different shapes availablable ranging from rectangular type, rod types (for use in pen like holders) to specilized shapes (animals, birds etc). With the rectangular style being used to erase large areas or complete sentences, with the thinner types being used where more control and precision is required.
Some erasers are made of hard materials (pumice) to aid in the removal pencil marks, but these can leave scuffs or even rub holes through paper.
As we all know erasers leave some mess around when used, so choose an eraser that leaves large pieces of debris which will make for easier cleanup. Note: The type of eraser used should be gentle on the paper with little smudging and be easy to hold
There is a good article at Jetpens about erasers which has allowed me to give a basic overview of the eraser types available.