If you win a fight, you will usually see a few screens like this:
Other than the satisfaction that comes from defeating your foes, you will also find new equipment for your Raiders! The left side of the screen will list the Raiders for whom the item is an upgrade and their current morale. Raiders are a fickle lot, and will only accept items with an equal or higher Gear Score to what they currently have. The item that dropped will be displayed on the upper right side of the screen. If you select a Raider, what they are currently using will be displayed on the bottom right side of the screen for you to compare the two items.
You can hover over a raider to see a tool tip showing information on his or her personality.
When you decide which Raider you want to give an item to, simply click the Confirm button (or press the Enter key). Note that those who do not get the item may be upset at being passed over. In addition to items, you will also find gold that can be used to purchase things from the store back in town. If you do not wish to give anyone the item, you can choose to disenchant it for additional by pressing the Disenchant Loot button, which will avoid any hard feelings over loot distribution.