Do Skyrim mods work with the Special Edition?

No, the mods made for 32-bit Skyrim (original) will not work for the 64-bit Skyrim Special Edition.

See here:

Skyrim Special Edition makes the jump from 32-bit to 64-bit, rendering the original mods useless.

Another issue is that there is no Script Extender for Skyrim SE at the moment, but it is currently being worked on.

It's also worth noting that modded original Skyrim files will not work in Skyrim SE. To transfer a savefile from original Skyrim to Skyrim SE it will have to be unmodded.

Some mods do work fine as-is while others need to be slightly modified, some remade completely. It all depends on what the mod in question does and how it does it. SKSE and anything relying and custom meshes are a few things that don't work out of the box.

I wouldn't recommend using mods made for oldrim in SSE unless the mod has been tested to work properly or you're familiar with creating mods for the game yourself.