The quintessential Medieval bow, the longbow is most famous for its use in the Hundred Years War (1337-1453). Made of a solid piece of wood, usually yew, a bow could be made within a few hours, and used for years. Lightweight and portable, this is a reasonable weapon of choice for a traveller, especially one who prefers to hunt for their own sustenance. The longbow is typically about the same height as the bearer, and customized to their size and strength.
Pros: easy to make; portable; reasonable maintenance; excellent for hunting; not immediately threatening; implies good marksmanship and considered a useful skill, which can be hired out; easily learned (although requires a lot of practice); a reasonable weapon for almost anybody, from a peasant to a nobleman; can be customized in all sorts of awesome/fanciful ways without significantly reducing effectiveness; a very good excuse to have your character wear cool looking bracers (basically forearm armour, usually leather, to protect the inside of the arm while shooting).
Cons: not easily repaired if actual bow is broken; extremely difficult to use on horseback (thus a better choice for a traveller on foot); requires a supply of arrows to be of use, which must be replenished over time; not effective as a short-range weapon (easily remedied by carrying a secondary weapon); may be subject to problems due to changes in heat and humidity; exceedingly difficult to conceal on one’s person.
Basic longbow diagram.
An example of the aforementioned bracer.