The benefit of this fancy micro is to 'run and gun'. Marines have a set attack speed, meaning if they are standing still they will shoot, stand still, then shoot again.

So by moving between shots you can constantly be moving away from melee attackers you can kite them (prevent them from hitting you therefore taking far less damage) while still doing the maximum amount of damage (because you shoot every time its off cool down). This becomes even more effective when you mix in mauraders with their concussive (slowing) shells.

To answer your question, this can be extended to ANY ranged unit in the game. It is very common with hellions, thors, marines, and mauraders with Terran

To do this micro, right click away from the enemy unit (ordering a moving command), press the A (attack move command) and left click (this will cause your marines to shoot anything within range) then order another move command; repeat.

Hope this helps!

So you've talked about two completely different strategies which are handled at completely different times in completely different ways. I've gotten tired of trying to explain to people when they have different concepts or the same concept so I'm just going to give you two answers right here.

Bunker Rush

I had the pleasure of watching Fruitdealer vs Boxer at Blizzcon 2010 and one of my favorite parts was game 2 when Boxer Bunker rushed Fruidealer at the start of the match. To his credit Fruitdealer recovered quickly and went on to win that game, but it illustrates what the goal and the strengths are of a bunker rush and how you have to respond

The Goal of a bunker rush is to punish a fast expanding Zerg. Like any other rush it makes a large economic sacrifice to try to devastate the opponent's economy by a larger amount. Unlike 7 Pool play, a Bunker Rush isn't designed to end the match, but rather deny the Zerg an early expansion. By denying that expansion and then going on to expand quickly the Terran player can put himself in a dominant position. Usually this is a debt that cannot be recovered from.

Also unlike a 7 Pool a bunker rush isn't an All-in. Instead, the Terran player usually only makes a handful of Marines and attempts to salvage his bunkers after he has denied the early expansion. This allows the Terran to quickly recoupe his expenses and expand. Because it is a low cost Rush it is a favorite tactic on small maps by many high level Terran players.

Dealing with a Bunker rush usually means two things: protecting your expansion, and not building too many Zerglings. If you over commit to Zerglings, even if the Terran doesn't kill a single Drone he has denied you a large number of Drones because those Zerglings could have been Drones. At the same time if you don't build enough Zerglings he'll easily destroy your expansion and leave you at an economic loss.

Responding usually means trying to destroy the bunkers before they are built. Pull 2-5 Drones off the line and try to attack the building SCVs. Because double Bunker rush makes an air tight wall this can mean you don't kill the SCVs before the bunker is built. Additionally, you'll need an earlier Spawning Pool than you expected, so as soon as you see the bunkers go down, build that Spawning Pool. If you have a Drone at the expansion its a good idea to get a Spine Crawler up to prevent it from being taken down.

Ultimately, if you can't defend that expansion you need to let it go and build sufficient forces to break out. This is a large set back, but not so large that you can't recover from it.

Marine Congo Line

An early Marine Rush is very different from a bunker rush as it is designed to be All-in play to end the game early. Instead of denying your expansion and trying to out Macro you, the Terran player is trying to force the issue early. Usually this will mean 2-3 Barracks (without any add-ons) and rallying his forces directly to your ramp.

The Marine is one of the most powerful units in the game. Like the Zerglings (and to a lesser extent the Zealot) the Marine represents very cost effective dps. Unlike the Zergling and Zealot, Marines use range to further take advantage of their cost effectiveness. This is often times why people say that the Marine is one of the few units that gets better as the game progresses. As a result, when dealing with the Marine you want to make sure you have a large enough force of Drones+Zerglings before you engage. By sending your units in wave the compounded damage provided by the Marine's range is going to punish you much more than if you were fighting Zealots.

To respond to these Marine Congo Lines, you're going to have to produce a large number of Zerglings and get a good surround. Wait to engage your opponent until the last possible second. The longer you wait the more units and resources you have, so don't engage until you have to. Preferably you'll want to engage on creep and move to surround instead of a-moving. This should trap the Marines and allow you to kill them off. If you're short on Zerglings, do not hesitate to include Drones. They do a fair amount of damage, and this is very much an all-in rush.

Why You're Still Losing

There is a reason you don't see strategies like these at the Bronze level, or even Platinum. These strategies trade very heavily on out playing your opponent by having better execution. Ever notice how Idra sends his drone to build a Spawning Pool before he has 200 minerals? Or how Fruitdealer seems to be able to produce 20 Zerglings at the last second? These players have honed their strategies and removed all the gaps in their build. This skill is often called "Execution," because it is the ability to execute your strategy flawlessly. It usually means little things like: Getting a good Split at the beginning; rallying new drones to under saturated mineral patches; sending out a Drone to build before you have the money for that building; never having 3 Larva at once; having the economy to throw together large forces on the fly. All these things are what constitutes having good execution.

There is a reason Day9 says

I could go just about any strategy and get into Diamond just because I have good execution.

Marine and Bunker Rushes trade on having these very tight, very optimized, builds against an opponent who doesn't, and executing perfectly. Until your execution is on par you are going to get punished by these builds.