You can only apply a poison to an equipped weapon: it'll tell you that if you try to use it without having a weapon equipped.
And you can only apply poison to one of your weapons: if you try to re-apply a poison after applying it to one weapon, it'll say you can't do it. However if you un-equip the weapon in your right hand and equip the weapon that is in your left hand to the right you can then poison it and then equip both weapons with poison.
You can, in fact, poison bows. Just make sure to equip it before applying the poison. This affects the first arrow shot through the bow: after one shot, the poison on the bow goes away. As a corollary to this, you can't poison arrows: Skyrim will complain that you don't have a weapon equipped if you try to use a poison with just arrows equipped.
And as BeatMe notes, with the Poisoned perk, you can reverse pickpocket marks to poison them stealthily. As long as the pickpocket is successful, this doesn't aggro the mark; in many cases, it'll outright kill them.