A disaster pod is likely a container of essentials for use by Colonial Fleet crew members for use during an emergency.
The contents of the pod are never disclosed, but it is likely that it contains food rations, first aid kits, basic medications, and repair and rescue tools, as well as serving as a pressurized emergency escape enclosure when emptied, similar to what sick bay becomes if Galactica is lost.
This container can then becomes a small unpowered ship with docking capabilities allowing it entrance on almost any ship, allowing people to be evacuated from ships where the normal airlock is damaged and unable to be used, this pod can be temporarily welded to almost any ship surface, allowing a small crew to escape into it and seal it behind them before releasing it so they can then be towed to a hangar deck and brought back inside a pressurized vessel
Captain Lee Adama asks Colonel Saul Tigh to issue two disaster pods to aid the civilians aboard a caravan of passenger, cargo and utility ships gathered by new Colonial President Laura Roslin.
Tigh grants the request mostly out of favoritism to Commander Adama's last surviving son, believed earlier to be dead, but insists that they do not use Galactica crew to obtain or distribute them and at the understanding that Apollo would then rejoin Galactica's airgroup as senior pilot and CAG (TRS: "Miniseries, Night 2").