Last-hitting is a technique where you (or a creep under your control) get the final blow on a neutral creep, enemy lane creep, enemy hero or building. The hero that dealt the killing blow to the enemy unit will be awarded with gold and experience.
For the exact amount awarded to the last-hitter and assist-gold refer to gold.
When trying to control Equilibrium on a lane you should always consider that by getting the last hit on any enemy lane creep you will naturally push the lane, since you are an additional damage source on the enemy wave. You can counteract this by attacking and denying your lane creeps or by Pulling.
Auto-attacking refers to the technique of simply letting your hero attack at will. This keeps friendly units alive, and helps when pushing, but it is much more difficult to last hit. In the late game, however, heroes often do enough damage per second to simply auto-attack and last hit. Auto-attacking can be useful in any case if you wish to push a lane forward, since it kills enemy creeps faster. This is especially useful for killing towers. In the early game, however, pushing towards an enemy tower is considered dangerous, as it leaves you far from your own towers and makes it much easier for enemies to gank you.
Denying is the act of preventing enemy Heroes from getting the last hit on a friendly unit by last hitting the unit yourself. All allied units can be denied once they fall below a certain percentage of health: creeps and siege units at 50%, heroes at 25% (only under certain debuffs), and towers at 10%. Illusion and summoned units act as creeps in these instances. This prevents your enemy from gaining any gold and a portion of experience from the kill.
Attack animations and projectile speeds
It is important not to push your lane too much, but try to keep the lane creeps fighting in approximately the middle of the lane so you never are too far from the safety of your tower. Take care not to attack the enemy team's ranged creeps as though they are squishy and easy gold they help push the enemy creepwave towards your tower. Creep equilibrium is generally maintained by balancing last-hitting and denying, and on occasions attacking your own creeps before the deny point or attacking the enemy creeps to push. This technique is also known as static farming. Creep pulling can also help maintain equilibrium, by resetting the equilibrium to the normal meeting-point. Take care the whole creepwave dies in the jungle though.
This is always situational though being pushed might be a bit safer but if the enemy lane is able to slowly chip away at your tower you should push harder.
Creep Blocking is a tactic that makes the equilibrium of the lane lean towards your side. This is accomplished by physically blocking your creeps' path with your hero so they take longer to reach the middle of the lane. Since your creeps move more slowly, they will meet closer to your tower, making it safer for you to farm, and more risky for the enemy.
Creep blocking is generally most useful for the first creep wave at the start of the game. In the middle lane, being able to stand on your side of the river - not in the river - to last hit is valuable. The high ground allows you to see your opponent's side of the river, and for ranged heroes avoids the 25% miss chance of last hitting uphill.
For the "long" or "safe" lane, creeps sprint until about halfway between the Tier 1 and 2 towers. This makes it much harder to creep block all the way from the barracks.
Pushing is the act of damaging enemy creeps through auto-attacks or through abilities. Pushing is used to move the creepwave towards the enemy tower, or simply away from the allied towers. Late-game, pushing is used to move creepwaves towards the enemy base to win the game.
Some heroes have abilities that facilitate expedient pushing, such as Leshrac's Diabolic Edict, Techies' Proximity Mines or Pugna's Nether Blast, which can damage multiple creeps and towers at the same time.
Pulling is the act of redirecting route of your lane creeps towards a neutral camp (usually the easy and hard camps, as they are closest to the lane). This has several aspects. Pulling, assuming you don't get a double wave, resets the creep equilibrium, denies XP (and gold) to your lane opponents and gives you additional XP and gold. Pulling creeps is not recommended when enemy creeps are about to push, as it will hurt your tower or carry (tanking creeps) more.
It is also a good habit to stack easy camp before pulling, because only 1 stack is usually not enough to clear the creep wave and you'll end up having so called "double wave", which will eventually make you end up by the enemy's tower. You can also repull again from easy camp towards medium/hard camp, using the same technique when neutral creeps are about to die, to prevent double waving. This is known as connecting the pull, double pulling, or chain pulling.
Pulling is usually done by a support, while carry should stay at the lane and farm lane creeps (which gives more gold/exp). It gives the support quite a good chunk of gold/exp, without leeching it from your carry. A distinct drawback being it leaves your carry alone in lane and more vulnerable to all ins from enemy heroes.
Roles and last-hitting
Depending on your role in your team you should focus on different things.
Carries should focus most of their attention on getting last hits.
Supports should focus on denying creeps and harassing enemy Heroes so that their carries can last hit without being harassed. Do not throw away the last hitting entirely as a support, because there are some last hits which are impossible for a carry to get (for example, the creep is being killed faster than a carry could possibly get in the attack range).
Supports can also creep stack to help their carry farm.