It varied from battle to battle.

As a general summary,

1. Archers need to be protected on the battlefield, by field fortifications or cavalry or other infantry forces, or by their own mobility (e.g., mounted archers).

2. Archers are effective against unarmoured opponents within range, including cavalry with unarmoured horses.

3. Archers can be effective against armoured opponents at short ranges. Generally, the thicker more protective parts of armour will stop arrows; this typically means breastplates and other torso armour, and helmets (or at least, helmet fronts). Arrows do not go through breastplates like in the movies! However, much of the rest of the body is much more lightly armoured (light mail, textile armour, thin plate (often under 1mm)). Such light armour can be penetrated.

The effectiveness of archery (especially against armour) is a popular question. A brief recent discussion is https://www.reddit.com/r/AskHistorians/comments/8dhbdx/how_effective_was_a_knights_armor_against_arrows/

A much more detailed discussion of armour vs arrows is https://www.reddit.com/r/AskHistorians/comments/7mu6z2/in_bernard_cornwells_agincourt_a_character/ with responses by, among others: /u/Hergrim /u/MI13 /u/WARitter

For an update by /u/Hergrim see https://www.reddit.com/r/AskHistorians/comments/7y6spm/saturday_showcase_february_17_2018/dufh228/

For recent discussion of mounted archery, and penetration of mail and lamellar armour: https://www.reddit.com/r/AskHistorians/comments/8egpff/how_effective_were_horse_archers_at_penetrating/

Getting arrows through armour is only part of the story. Archers can contribute to success in battle in ways other than just killing/wounding the enemy:

1. Archers can keep enemy archers (and other enemy missile troops, and sometimes artillery) out of effective range. If the archers outrange the enemy missile troops (e.g., Chinese crossbowmen vs mounted archers), they can render the enemy missile troops completely ineffective. If they do not outrange them, they can still keep the enemy troops at ranges beyond their optimum range, making them less effective, especially against armoured soldiers. Roman archers keeping the Parthian horse archers at longer range at the Battle of Carrhae (53BC) would have contributed to the Romans being able to withstand the Parthian archery for so long.

2. Archers can encourage attackers to bunch up, by shooting into the flanks of the attacking force (Crecy, Agincourt, Halidon Hill).

3. Archers can force an enemy to attack in less-than-desirable circumstances, by making the attack look like a better alternative than standing in formation while being shot by arrows (Neville's Cross, 1346).

Generally, archers will be most effective when shooting at unarmoured enemy troops who can't move effectively (e.g., unarmoured infantry forced to stay in formation to withstand potential attack by enemy cavalry (e.g., Falkirk, 1298)).

Where the enemy (or at least the front ranks of the enemy) were sufficiently armoured, and the archers did not have the opportunity to shoot at very short range, archers could be ineffective, or at least have little effect (e.g., Flodden, 1513).

Where archers were unprotected, and enemy troops could reach them, they could be badly defeated (e.g., Stirling Bridge, 1297, and Patay, 1429, where enemy cavalry defeated unprotected English archers). Note that archers alone could rarely stop an enemy unaided. At Halidon Hill, 1333, the Scottish infantry attack, advancing uphill into heavy shooting by the English archers, reached the English lines (where they were stopped by English armoured infantry (dismounted men-at-arms), while still being shot at by the English archers at short range in the flanks, which led to rapid collapse).

One circumstance where archers were well-protected and played a very important role in battle is sieges. Archers would force enemy artillery to stay out of archery range, and would force much of the preparation for attack to be carried out out of archery range. Archery would also be used to attempt to stop preparations for attack that couldn't be carried out out of range (filling in moats, using battering rams, etc.).

I won't speak about archers across the board. However, English longbowmen were highly effective, especially during the Hundred Years' War. They truly established their place in the English military during this time, but they had been used prior to that.

A longbow could fire at a distance of up to 270 yards and up to 75-80 with any accuracy. A trained archer could fire 10 arrows in a minute, and they could lay down fire for 3-5 minutes depending on how many arrows each person was stocked with.

All soldiers aren't wearing plate mail; so arrows would be effective on lightly armored opponents, like the typical infantry and enemy archers/crossbows. Special arrowheads were also designed that could actually penetrate through the rings in chainmail. Blunt arrows could also cause damage to helmets and the heads inside. Your helmet isn't going to help much, if you have a concussion and your visor is crushed into to where you can barely see.

On top of that, the heavily armored knights would often be riding much more lightly armored horses. These horses would go down, severely limiting the mobility of a heavily armored knight. Arrows could also effectively pierce any joints or lightly armored areas of a knight's armor.

Longbows were so important to the English army that taxation in the form of yew bow staves was paid on trade goods in the 15th century.

The Battle of Crécy in 1346 is probably the best example of the dominance of an English longbow. An outnumbered English army established a line waiting for the French force to arrive. The French were angered that the English were able to cross the Somme and rushed to meet them.

The French advance forces rolled up and decided to attack, despite needing to wait for the rest of the army. They advance with Genoese crossbowmen at the front of the line. The Genoese started picking their way through the obstacles the English had placed on the approach. They attempted to lob vollies into the English line, but weather had weakened their bowstrings making them ineffective.

English longbowmen flanked the English line and were lobbing shots into the approaching French army. With the failure of their crossbows and getting dropped by English archers, they split ranks and fled back into the approaching French line causing chaos.

Arrows continued to pour into the French army, and dead horses and knights began to block the path of retreat. The longbowmen were able to do most of the work in repelling sixteen French charges. By the time all had been said and done, the English had lost around 100-300 men; while the French had lost an estimated 13,000 (including several important nobles) with some estimates going up to 30,000.

As you can see, there are times in which the longbow was used highly effectively. And there are other instances of these same tactics working. Prince Edward the Black (same English leader as in the Battle of Crécy) used a similar strategy during the Battle of Poitiers in 1356 which allowed him to once again overcome a superior force with few casualties. In his letter to his father he reports fourty deaths compared to the French's 2,000 (it was probably higher but not by a whole lot).

King Henry V at the Battle of Agincourt in 1415 was yet another situation in which longbows rained havoc on the French. Severely outnumbered and unsupplied, Henry V lined his infantry between two woods that would funnel in the superior French numbers. In the woods he placed two longbow units flanking the infantry.

He baited the French to attack, but their hesitance allowed Henry to gather his forces into a better position. The longbows launched waves of arrows into the French cavalry. The cavalry launched a disorganized charge and were cut down, pretty much just making the French attack route muddy and difficult to traverse. The rest of the attacking French were cut down by arrows or exhausted from the mud. They engaged in some skirmishes but were pushed back. The casualties at the Battle of Agincourt are a little hazy, but there were more captured French nobility than there were English casualties.

As you can see, arrows are effective against mounted (which indirectly affects heavy armor) and lightly armored units. They can also cause confusion in the ranks and strike from a distance, forcing retreats before the melee even begins. So yes, I would say that archers were effective. Of course they had their weaknesses; but when used right, they can be deadly.