“Rotamers”, rotational isomers, are a consensus set of low-energy sidechain conformations. In other words, they’re a set of possible sidechain configurations which captures the vast majority of physically likely positionings.
Packing is an algorithm which chooses a low-energy set of rotamers for a given backbone (where some of the rotamers might be fixed into specific positions). This lets one configuration of lysine become a different configuration of lysine.
Design uses the same algorithm as packing, more or less, except the chemical identity of the residues can change. This lets lysine become valine, etc.
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