I started using a company called newtone a few years ago. Mainly because they are local and priced well, but i was also hearing great things about the tone and string life so it was worth a punt.
http://www.newtonestrings.com/mechanics.htm
they offer as standard:
Hex cores - a bright punch sound which is expected from electric guitar strings
Round cores - extra warmth with a little less harshness, particularly usefull on naturally bright guitars. The Round coes also help reduce the tension of the wound strings.
Archtop - A Double wrap of Nickel over a Round core, warm and melow without the string squeal of single wrap strings.
Stainless steel - extra brightness you would expect from Stainless Steel.
Drop D - wound on Hexagonal cores, but featuring a thicker core on the 6th string to allow tuning down to D without loosing the definition of the note due to lack of tension.
plus many more selections for bass, acoustic and other instruments plus they will do custom sets made to order using any of the techniques above. I really like tehir bass strings made with hex core and a double wrap = the outer wrap is so fine they feel comfy like flatwounds but dont sound like mud
I think most string makers offer a choice of gauge and maybe material - but not much else
So what can affect the tone and feel of a string (not including guitar factors like magnetic pull and scale length etc..:
* core material
* core size
* core shape
* Outer wrap material
* outer wrap size
* outer wrap shape
* core to wrap ratio
* lasty but probably most importantly - age!