How can you teleport an entity relative to another entity?

Solution 1:

The current answers are all outdated. Now there's the /teleport (archive) command, which has none of the side effects of the current answers:

/teleport @e[type=armor_stand] ~ ~-8 ~

/teleport teleports relative to the executor instead of relative to the target, which makes it consistent which other commands.

In 1.13+ you can also write tp instead of teleport, because it behaves the same.