What bow should I use to maximize my ranged damage?

To answer this question, it entirely depends on what stats you are focusing on. Scaling plays a big part when choosing the right weapon, in any situation.

What scaling does:

Scaling boosts the damage that you do with any particular weapon, depending on which stat(s) and which level of scaling are defined (S, A, B, C, D, E) as @slow_excellence pointed out; S being the best, E being the worst.

This picture shows a boost from strength scaling (shown by the blue numbers):

How scaling affects damage boost:

• E: one point every 3 levels.
• D: 2 points every 3 levels until 30, then 1 point per level till 40, then 4 points per 10 levels.
• C: One point per level until level 30, then 1.5 points until level 40, then 0.5 points until level 50, then 0.3 points until level 60.
• B: 2 points per level until 30, then 3 points until level 40, then 1 point per level until 50, then 0.5 points every level after.
• A: 2.5 points per level until 30, then 3.5 points until level 40, then 1.5 points until level 50, then 0.6 points every level after.
• S: barely higher than A, but still enough to make a difference.

Bows and their scaling stats:

• Short Bow: STR - D / DEX - A
• Long Bow: STR - D / DEX - A
• Composite Bow: STR - C / DEX - C
• Sea Bow: STR - C / DEX - C
• Dragonrider Bow: STR - B / DEX - D/ INT - C
• Bell Keeper Bow: STR - C / DEX - B
• Bow of Want: STR - C / DEX - A / LIGHTNING - C
• Hunter's Blackbow: STR - E / DEX - S

Greatbows:

• Alonne Greatbow: STR - C / DEX - C
• Dragonslayer Greatbow: STR - B / DEX - C

Also important to note that you cannot move while firing a greatbow, unlike a regular bow. This can affect your play style greatly.

It all depends on how you are attempting to scale. If you're going for a strength bow,use the Dragonrider bow,it has a B scaling on strength. If you are aiming for dex,use the hunter's blackbow. I have recently started up an archer and I've tested the differentiation of the bows. I hate greatbows mostly because you can't move with em',making it a gamble if you miss your shot in PvP.

I run 45 dex and 40 str and the hunters black bow+10 has the highest damage excluding great bows. Next highest damage is the longbow+10 then the Sea Bow +10 which is actually my favorite because it has the highest rage out of ANY bow in the game. Also the hunters blackbow has the fastest fire rate as well. So it is definitely worth trying a few out and seeing what you like for yourself as opposed to asking other people what YOU should do with your own build. That's how I figured out what I liked. I just did a bow only run and planned on using the longbow but ended up using the sea bow for primary and fire short bow as a backup. Hope this helped :) ps:don't forget to buy poison arrows