To "screen a march" is to deploy another unit to cover the movement of the first unit. The screening unit would generally be lighter and more mobile than the unit they are covering. Their purpose is to impede an enemy assault on the moving unit. If an enemy were to launch an assault on the unprotected advancing force, that force might not have time to orient themselves to face the attack. However, if a cautious commander assigned a screening unit of skirmishers to guard the advancing force's flanks, then the enemy will have to deal with those men before attacking the main body. The skirmishers won't be able to stand up to heavier units, but they will be able to delay the enemy or disrupt his assault.

Screening forces are a standard military practice and remain in use by modern militaries, albeit in a somewhat modified form than the way the Romans understood it. You can see the principle of "screening" an advance down to the fireteam level of combat in modern small-unit tactics: http://en.wikipedia.org/wiki/Bounding_overwatch. Two fireteams screen each other in turn so that they can "leapfrog" an advance up the battlefield.

Screening forces were less a sign of organizational prowess and more a sign of having a military doctrine that taught this method. Screens were actually uncommon in the ancient world, and as a result armies sometimes wandered in circles for days looking for each other before major battles. In the most famous example I can think of Alexander the Great passed the Persians up on the way to Gaugamela and had to turn around to close with them.

The use of screening forces has gotten more popular throughout history, because it is a very effective way of ensuring you don't get surprised, and that you have enough time after making contact to orient your forces for best effect.

Wellington was a major innovator with his use of rifle skirmishers as screening forces. The rifles were able to move quickly and engage at longer distances than regular infantry units (reg infantry fired massed volleys at close ranges, often 50 yards or less, rifles could engage the enemy hundreds of yards out). He would have them take point on marches, or the flanks, and when battle was joined he would place them on a flank. He used them defensively as well, when his army wintered the rifles took the most remote outposts on the line.

In WW2 the Germans added a major screening innovation along with their other groundbreaking innovations in the field of mobile warfare called reconnaissance in force. When moving large armored columns quickly good scouting intel is critical, the Germans learned fairly early on that small scouting parties were prone to getting lost, or worse, getting found and quickly overwhelmed by the enemy. They would send a larger unit to scout, one capable of surviving a fight and gathering and delivering information back to the commanders. This was adopted by US and NATO forces during the Cold war.

During the Cold War the NATO defensive posture in Europe called for a layered defense screened by armored cavalry units posted near the front. They were to fight a holding action, delaying an enemy advance while the heavier units behind them got ready. When advancing large units would be screened by armored cavalry regiments, they would locate and close with the enemy and allow heavier units to orient on them and engage in force.

One of the most famous actions of the 1991 Gulf War is the Battle of 73 Easting, a good example of this doctrine. An Armored Cavalry Regiment screening the US VII Corps ran into (literally) 2 Iraqi Republican Guards Brigades, one Armored and one Mechanized Infantry. The Armored Cavalry Regiment was a pretty heavy unit, with contained armor, artillery, and aviation (helicopter) units. They joined the battle, and were supported in it by heavier units of the US 3rd Armored Division.

As you can see screening units never fell out of practice, and are used today even with satellite and air imaging. The early concept of screens was to scout and keep an eye out for enemy units, it developed into creating a buffer force around your main body of troops allowing you time to orient on the enemy and establish your order of battle. With the advent of modern mobile warfare screening forces got heavier and more capable of fighting, and the modern sensibility on the march is to locate and engage with a security force and move heavy units in once contact is established.

If you are American the best examples can be found in the American Civil War. A Cavalry force would march along and in parallel to the main body (normally infantry) covering its flanks or at least the one facing the enemy body. In addition, not part of the screen really but part of the tactic, you would have similar Cavalry units moving ahead of you as scouts to pave the way and collect intelligence of the enemy force to determine what was in their front.

The mission of the so called Cavalry screen was to avoid detection and penetration of the enemy force to reconnoiter and gather intelligence by fighting them and keeping them at bay, thus the enemy force will be unable to determine if the movement was real or a distraction. The "screen" would prevent enemy forces to determine the strength of the main body, was it the whole army moving? only 1 division? in what order was it moving and what units were involved? A successful screen would place the enemy commander in a difficult position to make his next move. Should he move with his whole Army too in pursuit? if this was a fake movement he would move leaving his rear exposed chasing a ghost and leaving his lines behind unprotected, Is the enemy trying to lure him as far as possible to strike somewhere else? You can probably see the implications unfolding by themselves.

Worth mentioning that you can produce a screen with natural obstacles too, even today, remember that satellites and sometimes planes are not available to everyone and modest armies or unconventional forces would have to do whatever possible to conceal their movements. In line with my paragraphs above would be Lee's army march North to invade Pennsylvania prior to Gettysburg; he left Virginia marching North along the left side of the Blue Mountains using them as the screen to conceal his move having the advantage of having to protect the mountain passes only.

JEB Stuart commanded Lee's Cavalry and should have been providing the discussed screening but Lee sent him on another mission...but that is part of a different story.

Hoped it helped.