My favorite kind of answer to my favorite kind of question:
Some mods, like mods regarding high-res textures, are single-player mods that do not affect the communication with the server, and can therefore be used irrespective of server configuration.
Other mods, like aircraft mods, require server support in order to work.
In general, the description of any given mod will list if it's compatible with vanilla servers, if it only takes effect in SSP, or if it requires a modded server.
Edit: In response to comments; one piece of clarification:
Some mods function somewhat more like exploits based on the fact that the server allows a certain amount of client-side processing of things like falling. Mods that function by misrepresenting the effects of gravity, for instance, can function without server support, at least until the communication protocol is changed to handle all movement server-side, assuming that ever happens.