Upgrading weapons is a very important aspect of Dark Souls, and the process itself isn't as threatening as it may seem. Dak Souls veterans reading this probably already know where to get all the ore and which upgrades are best for which area, but I'm sure lots of folks here have troubles keeping up with all the divine weapon/titanite chunk shenanigans. I know that all instructions can be accessed via darksoulswiki.wikispaces.com, but I think it would be a good idea to create an easy to access thread explaining how to master the art of upgrading; more experienced players could also aid their more noobish brethren in the process.
To upgrade a weapon you first need titanite shards (can be bought from blacksmiths for 800 souls a pop) - blacksmiths can upgrade your weapons gradually using a certain number of them (1-3) and a small amount of souls.
Important note: After upgrading your normal weapon to +5 you have to visit a blacksmith to select the next upgrade path you want to follow. For example, if you want to get a divine weapon, you have to reinforce a normal weapon to +5, then visit a blacksmith and modify it to divine +1 (You can do this only with the help of a blacksmith); The weapon will then be available in the "reinforce" option, and until the next upgrade stage (in this case from divine +5 to divine +6 or occult +1) you will be able to reinforce it at a bonfire.
When you get your weapon to +5 you then have the access to various branching paths, as seen in this chart:
EDIT:Props to @Immuniity for providing the updated version of the chart, thanks a lot man!
(Bigger size: http://media.giantbomb.com/uploads/9/97252/1973752-dark_soul_weapon_info.jpg )
(Locations of various ores and upgrade specifics: http://darksoulswiki.wikispaces.com/Upgrades)
(Dark Souls wiki homepage : http://darksoulswiki.wikispaces.com/)