Under what conditions can a C corp increase the total number of shares authorized?
Board approval? Majority shareholder approval? Unanimous shareholder approval?
The board typically issues authorized shares, but additional shares are generally authorized for issuance by the current shareholders by majority approval.
The precise answer would depend more on the current status of the C corp and how shares have been issued and sold.
If this is still the "3 guys in the basement" stage, you pretty much will just file paperwork with the state, update your meeting minutes, and it's not a big undertaking.
If you have actual investors that have purchased shares you will most likely want to conduct a formal vote, with records of the vote, and then proceed to authorize the new shares and file the paperwork mentioned above.
This is done by amending your articles (or certificate) of incorporation. This generally involves both board action (to approve the amendment and recommend it to the stockholders) and stockholder action (to actually approve the amendment.) Once the amendment is approved, it's filed with the Secretary of State, and voila -- you now have more shares authorized.
If your board and shareholders are the same people, folks often do this in one fell swoop with a joint board/stockholder resolution. Whether that works for you will depend on where you're incorporated.
If you haven't issued shares yet, then you don't need shareholder approval.
In general, you do NOT need all stockholders to go along with any action, including amending your articles. But, they all have a right to know about the vote.