Armor is used to provide protection against weapons fire during ground combat. Armor in addition to providing protection also gives certain attributes buffs.
The descriptions for many types of armor say that the armor is "bound" to you when used. This means that if either you or one of your bridge officers wears the armor you will not be able to sell it on the Exchange to other players. (You will still be able to get rid of with the Replicator and earn some Energy Credits though.) If an armor seems especially interesting, but does not really suit you or is below your current level, you could give it to one of your bridge officers, but you might want to consider just carrying the thing around unworn till you can get to the Exchange and try to sell it.
You can hide any body armor you might be wearing. Hit the "U" key to bring up your character's status and equipped items, right click on your armor and there will be an option to hide it. You will then appear to be wearing your normal starfleet uniform, even though the armor will continue to protect you. Hiding armor is especially nice in social areas like Space Dock because it looks a little odd to be wearing full battle gear while chilling out in the bar. Plus you can show off the custom uniform colors you chose to other players! (Why bother choosing your own uniform if no-one will ever get to see it after you've picked up your first armor?)