Shields would have been ubiquitous in all medieval armies, at all levels. The only time when they start to become less important is around the late 1500s, when firearms are beginning to shift the nature of warfare towards ranged combat and full suits of advanced plate armour make it obsolete for heavy cavalry.

At the simplest level, a shield is piece of material between you and a projectile. Without it, even the simplest ranged weapon (say, a sling) could kill you. It goes without saying that it protected from melee weapons too, and it could be used with very little training and benefitted from a rather low price, affordable by many combatants.
Within the logic of armies where most of the infantry is rather poor and untrained, a shield is a convenient substitute for body armour. This shield from the Gokstad burial (around 900 AD) is roughly 56 cm in diamater, most likely similar to those used before in Carolingian times and later, until the end of the century when we see the emergence of the so called "kite shield" as shown by the Bayeaux Tapestry. These shields, as far as reconstructive archaeology and reenactment tell us, are more than capable of defending the body from blows and projectiles.
Even in later times, as this hypothesis of drawing of the Battle of Campaldino (1289), provide us data to show us so.

Even knights and mounted fighters would employ shields, although there is a tendency for shields to shrink in size as armour keeps advancing, as shown in this XV century miniature of the Battle of Crécy (1346), where knights are depicted as they would have during the 1400s and not the time of battle. In any case, they are fully enclosed in plate armour and discarding the shield. Plus, the most basic tactic a unit can perform, even if it's composed of fighters with little training, is to close ranks behind a line of shields (without even mentioning purpose formations of heavy infantry with shields, suck the Roman testudo and Greek hoplite phalanx).

In order to get past a shield, several options would have been available, based on the circumstances of the combat taking place. In case of a 1v1 scenario, it would depend much on the type of weapon the aggressor would be using. Assuming it would be a polearm (the other ubiquitous weapon on the battlefield, especially spears), the combat would result in attempts to thrust at an opening in the opponent's defence, maybe using overhead or underhand thrusts, maybe aiming for the legs or entering a bind with the opponent's weapon.
In case of a hand weapon (say, an axe or mace), the dynamic of the encounter would change again as grappling could be a very useful option, with the possibility of using one's shield to apply leverage or blocking the other's shield to create an opening. This is what shield bosses (the metal cup in middle of the Gokstad shield) may have used for in an offensive way.
Finally, if you have a two handed polearm (a pollaxe, a halberd, a Dane axe), the considerable force you could exert while connecting an attack may result in bruises and concussions to the arm holding the shield, possibly damaging the shield or exploiting an opening so created.

I wish to point out that shields of various sizes were used in different ways, depending also on the offensive implement they were paired with.