Words for Adding Things
Note that there are actually many more suru-verb kanji combinations that mean things similar to "adding" or "growing," so I've only covered the two mentioned here.
This is used only for adding two things of the same type.
Example: なべに水を足す "Add water to the pot"
Like 足す, this is also used for adding two things of the same type, but it is also used when adding things of different types.
Example: カレーに薬味を加える "Add condiments to the curry"
増える【ふえる】 / 増やす【ふやす】 / 増す【ます】
Unlike 加える and 足す, these do not have a meaning of "adding," per se.
増す is generally used when increasing a degree, level, or extent. It can be used with を or が. When used for volume or amount, it adds a moderately formal air.
Example: 悲しみが増す "Increase [the degree of] sadness"
増やす is simply "to increase the amount," and is generally used with を.
Example: 楽しみを増やす "Increase [the amount of] pleasure"
増える is "to be increased" or "to grow" and is generally used with が. They are both used to describe volumes rather than degrees. Note:
Example: お客さまが増えた "[The number of] customers has increased"
Much the same as 増える and 増やす, this means "to increase," much the same as the English verb. The size of the increase does not matter, but it can only be used with quantities or volumes, not degrees.
Example: 体重が五キロ増加する "Weight increased by 5kg"
I've also included istrasci's suggestions, which rather than simply meaning "to add" all mean more along the lines of "to supplement."
Used to add in something after-the-fact which was insufficient or lacking.
Example: 炭を付け足す "Add charcoal (presumably to a dwindling fire)"
Used to add in something after-the-fact which is necessary. Very similar to 付け足す.
Example: 名前をつけ加える "Add a name to the list"
付加 is used when adding something after-the-fact which is different than the original. Very similar to 付け加える.
Example: 条件をもう一つ付加する "Add one more condition"