So I'm not asking about shooting into the enemy formation without specific target but targeting a single individual with a reasonable chance of success.
I'm going to amend the question to fill in some gaps.
From how far away could an experienced and skilled archer hit and seriously wound a man?
This removes things like target arrows and light draw bows which are more accurate but less effective.
You didn't specify a particular society or type of bow leaving a pretty broad swath of bows and styles to consider. Despite the "middle ages" and "ancient-history" tags, I'm going to consider both historical and modern bows. Modern advances like compound bows (developed in the 1960s), counter weights, bow sights and modern materials are helpful and will serve to push the range up.
I will also assume the person isn't wearing more than normal clothing. If you want to get into penetrating armor with an arrow, that's another question. Most soldiers on a medieval battlefield would be wearing padded or leather armor at most.
I've done everything in metric. As a crib for the Imperialists, 1 meter is about 1 yard.
Getting an upper bound
First, because people have been throwing around answers like 350m and 500m (?!) let's go for an upper bound: Olympic archers are the best we'll ever do. They're the best trained, under perfect conditions, and they're using extremely accurate, but combat ineffective, Olympic bows. These have a fairly light draw, 20kg, and very light, very fast, and very narrow arrows. I'm not saying you'd want to get hit by an Olympic archer, it would hurt a lot, but they'd be ineffective in combat. Point is, this is the best we can possibly do.
An Olympic archer can reliably put an arrow into a 12cm ring at 70m. A man sized target is about 3 times as large. Apparent target size is directly proportional to distance. Ignoring the problems of wind and arrow drop we can put our upper bound at about 200m.
Now what can a more average shooter do in combat conditions (noise, smoke, wind, exhaustion, moving target, getting shot at...) and firing lethal arrows. For that let's look at people who already do this bow hunters. Bow hunters have their own competitions called Field Archery and from this we can glean a reasonable expectation of accuracy. Wikipedia claims bow hunters will fire at about 15m, with the maximum being about 40m, but it's based on one graph. A bow hunter wants to make sure they hit on the first try, else they're scare their game, and the game will be moving, even if not very fast, so this is probably the most realistic number.
Field Archery Competitions
We can look at the rules of International Field Archery for their expectations. Certain Field Archery competitions are closer to what you'd experience in combat, in particular "IFAA 3-D Hunting" uses animal sized targets placed at unmarked distances. The archer does not know how far away the target is. The maximum distance a target can be placed is 60 meters. This target is 23cm x 37cm wide approximating a man sized target.
Similarly, in the "Hunter Round", the maximum distance is 70m but firing at a 65cm target, twice the width of a man. Man-sized 35cm targets are fired at from ~30m.
The International Field Archery Association has a "Historical Bows" category which defines them as "based on the accepted design and usage during the period preceding the year 1900". They make no change to the targets for historical bow competition.
The effect of historical vs modern bows can be seen in the 2014 Tournament Results. The best scores with historical bows (AFHB and AMHB, Adult Male/Female Historical Bow) are routinely lower than the worst scores with modern bows. This could reflect the inaccuracy of historical bows, or it could reflect the smaller talent pool of historical bowman.
Based on all this, I can draw these conclusions about a human with a bow hitting a man sized target using a modern bow.
- The absolute maximum range by the best human with the best equipment is about 200m.
- A skilled, experienced archer can hit a stationary target reliably at 30 to 60m.
- The same skilled bowman will prefer 10 to 20m for a moving target.
These are numbers for a modern bow. They should be considered upper bounds for a historical archer who would not have access to modern technology to improve their aim.
There's no clear-cut definitive answer for this as most battles involved mass ranks of soldiers and not individual targets. Archers would be trained to shoot at groups of enemy and not individuals, at least not at any great range.
The English/Welsh Longbow of the early 15th century, had a range of up to 350 yds or so, but it was only considered effective at up to around 200 yds. This doesn't take into account armour worn by the enemy. For example a knight in full plate armour would probably be impervious to the Longbow at ranges of much above 50 yds, depending on the type of arrowhead used.
Shooting a warbow I'd expect an average archer to hit an individual man at about 50 yards, if not with the first arrow then probably with the second. Some will think this is an underestimate, but they probably don't shoot. The problem is one of judging the distance to get the correct elevation. I shoot field archery (rubber 3D animal targets at unknown ranges) with a primitive bow and a light arrow which would roughly equate to the trajectory of a warbow with a heavy arrow. The standing bear target is a good approximation to a man and is typically shot at about 40-60 yards. At that range it's a matter of putting the point of the arrow on it's nose and expecting it to drop into it's chest. Any slight wobble left or right will make you miss, now imagine holding 140 pounds of draw weight steady at full draw... No enemy is going to get closer than 40 yards without an arrow through him, but some of the long ranges suggested are fanciful, mind we all have the occasional lucky shot. Just for the record:-heavies bow I've shot 100# heaviest I've made 130#, longest shot 302 yards with a flight arrow from 100# Elm Warbow. (Google Delsbows or Bowyers Diary)
A composite bow, particularly a Mongolian style bow would easily out range an English war bow by a hundred meters, there are even claims of 500m shots. Thoe the question should really specify what the target is doing and wearing, if it's anything other than being stood in a meadow, in his underwear, and on a dead calm day I'd suggest the effective range of any bow, against a specific stationary human sized target, is no more than 70-80m. Though modern field archery is practised at 20-90m, so with a modern sighted composite bow, carbon fibre arrows, and a bit practise you'll get a few more yards.
Note: Compound / Recurve bows store and release energy far more efficiently, so for a given draw weight (Newtons of energy required to pull a string back) an arrow will leave a compound bow far faster than a longbow. Note the relative speeds of the compound bow and Longbow on these sites. You can use the numbers to determine range on any of the numerous online calculators.