There are several ways to generate gold in League of Legends:
Gold from minions: Gold can be earned by landing the killing blow on enemy team minions. The amount of gold earned is dependent on the game time. The longer a game of League of Legends goes on the more gold each minion is worth. Additionally, different minions are worth different amounts of gold.
Gold from monsters: Gold can be earned by landing the killing blow on neutral monsters. As the game progresses, monsters will yield steadily increasing amounts of gold. For instance, a golem monster may yield 50 gold at the start of the game, but over 100 later on. Certain kinds of monsters called epic monsters give gold globally, providing gold for the entire team regardless of which individual champion landed the killing blow.
Gold from champion kills and assists: Gold can be earned by killing enemy champions. The gold gain associated with a champion kill is determined by how many kills the slain champion has recently accumulated (champions that have killed many enemies without dying are worth more gold). A percentage of this gold is awarded to the champion who strikes the killing blow, and a percentage is then split amongst all the champions who assisted the kill (damaged the slain champion within 10 seconds of his death).
Gold from champion abilities: Some champions have abilities that add gold under certain specific circumstances. For instance, Ashe's ability Plentiful Bounty adds additional gold each time she scores a killing blow on a minion or champion, and Katarina's Voracity ability adds gold each time an enemy champion that she has recently damaged is slain.
Passive gold gain: The gold a champion earns without actively killing units is called passive gold gain. Every champion passively generates gold slowly over time. This gold gain does not begin until minions spawn at each base. Passive gold gain can also be the result of runes, summoner masteries, or gold generating items like a Philosopher’s Stone.
Selling items: Items that you find you no longer need can be sold back at the shop for 70% of their total cost. Items that generate passive gold gain can be sold back for 50% of their total cost.