The yew bow is perhaps the most traditional form of the English longbow, being a popular and powerful weapon throughout the Middle Ages. Yew wood is unusually good at holding and releasing energy -- that is, it's strong and springy -- and in England it was one of the best possible bows to be found at a reasonable price.
In England a war bow and a yew bow were almost synonymous, and England continued to use these bows long after other parts of Europe had moved towards the crossbow (there were, in fact, often laws in England banning the ownership of crossbows, to maintain the superiority of the long bow). The standard yew bow was 60-70 inches long, a self bow cut so as to consist of a layer of sapwood (the bow's back) and a layer of heartwood (the front). The string was made of hemp or flax. While estimates vary, they had draw weights of about 90–100 pounds-force, and could fire a 3-foot long arrow for up to 350 yards.
There is a fair amount of folklore tying yew trees to the church, and many of these try to integrate archery into the lore: yew trees were planted in graveyards because trees fertilized with corpses make the best bows; they were planted in churchyards to prevent archers from making bows to oppose the King’s men; trees intended for bow-making were planted in churchyards so they wouldn't be eaten by grazing animals. Most likely none of these are true, as English yew trees are not very good for making bows; the best trees were in mainland Europe.
I have not been able to find clear answers as to why European yews were better for bow staves. It may be that the European cultivars were simply genetically more free of knots; it may be that trees grew taller and more knot-free in alpine and Mediterranean climates; or it may be that the continued over-harvesting in the British Isles meant that there weren't enough mature yews to keep up with demand. I suspect the latter, as England produced a tremendous number of yew bows; they were in wide use by the mid-1200s, and by the mid-1300s importing staves was a constant necessity.
This was not only a matter for career archers; in 1242 Henry III made a law that all men owning land worth more than 40 shillings must own a long bow. In 1357 Edward III banned the export of bow staves, and 1363 he made weekly archery practice compulsory for all able-bodied men, and made archery competitions the official sport of the land, banning all lesser sports. In 1472 king Edward IV started a long tradition of requiring merchants to include bow staves in any ship importing goods; for many years specific quotas of bow staves were the required form of taxation for imported goods.
Meanwhile, the rest of the continent was using compound bows, made with a wooden core, layered with dried animal tendon, and faced horn, all plastered together with strong glues. These were harder to make, but worth the extra work if you lived in warmer climates, as yew would start to degrade and lose its springiness when the temperature approaches 100°F (38°C). On the other hand, tendon does not perform well when it becomes damp, a problem in the English isles. The use of yew made widespread production of quality bows much easier specifically in cooler climates that had trading ties to the Mediterranean. In practice, this meant that yew bows were in common use in England, parts of Germany and the Low Countries, and parts of France.
The decline of the English longbow started in 1450 at the Battle of Formigny, where the French used cannons to break the formations of English bowmen; this happened again at the Battle of Castillon in 1453, and the age of gunpowder slowly started to take over warfare. The yew bow was still in common use for decades after, and played a major role in the Wars of the Roses, but the Renaissance saw it fade to a marginal role.