Overview
Farming is doing nothing but killing a certain type of monster, Creeps and Neutrals in this case, for the express purpose of getting gold.
Farming, along with ganking and pushing, is one of the main aspects of any DotA game. Put simply, killing creeps and neutrals repeatedly is the most common and effective means of amassing gold for items, far surpassing that received from Hero killing and Tower destruction. Since items in turn often decide games, knowing when and how to farm is an essential skill for every player.
Farming Techniques
Arguably the most crucial skill in farming is the ability to Last Hit. Since players only receive gold if they deal the killing blow to a creep (or hero), waiting until the enemy unit's health is low before attacking or using a spell allows players to ensure that they receive the bounty. This technique is most frequently utilized early in the game, during which last hitting competes directly with Denying creeps, but should be practiced throughout the entire game to maximize farming efficiency.
While all heroes are capable of farming, some are clearly better than others thanks to their spell sets. Spells that deal damage to multiple units or in an Area of Effect, e.g. Pugna's Netherblast and Leshrac's Lightning Storm, can easily net the player several creep kills in a single wave if timed correctly. To maximize the effectiveness of such skills, the player should wait to use the skill until as many enemy creeps as possible are weakened to the point that they will die immediately to the spell; this is often achieved by having the hero attack enemy creeps that friendly creeps are not currently focusing on.
Skills and items that grant extra AoE damage are also highly valued in farming for their utility in quickly clearing entire creep waves. Cleave from Battle Fury or a skill (e.g. Magnus' Mighty Swing) is an excellent farming ability that becomes stronger as the game progresses and the hero's damage increases. Radiance is perhaps the best farming item in the game, granting a damage aura that often nets unintentional creep kills in addition to greatly increasing damage. Shiva's Guard deals damage to every unit in a large area; when coupled with another AoE spell, entire creep waves can be quickly destroyed.
Farming creeps is inherently negatively coupled to the process of pushing lanes. The ideal position to farm is next to an allied tower, close enough to retreat to but far enough away that the tower itself does not attack and steal enemy creep kills. If done carelessly, however, farming the same lane will cause the creeps to move away from allied towers and closer to enemy towers, which both slows down farming and renders the hero vulnerable to ambush. Maintaining the position of the creep line and not pushing is therefore crucial to safe and efficient farming. To preserve the equilibrium, a player should refrain from using damaging spells and instead kill solely through last hitting. Furthermore, friendly creeps must be denied whenever possible, or else the creep line will inevitably shift towards the enemy's side due to the player's attacks. While seemingly boring, this process is the most effective means of farming over a long period of time, and should be practiced by any heroes not actively involved in ganking.