I typically play Skyrim with a number of mods installed, and while I'm excited to try the Special Edition, there are a few mods that I'd much rather not play without. Can I install mods built for the normal Skyrim game with the Special Edition, or do I have to wait until mod authors write new versions before I can play with those mods?
No, the mods made for 32-bit Skyrim (original) will not work for the 64-bit Skyrim Special Edition.
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.
I know of 2 mods that work perfectly with the special edition. Better Dialogue control and Better Messagebox Controls. The 32bit skyrim page states the same if you visit their pages.
The "Best Answer" is actually kind of misleading and not entirely true. The short answer is no most will not, however certain mods will work, let me go into a little bit of detail. Any mod that uses scripts will require a script extender to work and currently a script extender for SE is not out so they will not work at all, but any mod that does not use scripts and is only an esp file will work just fine as long as you place it in the skyrim data folder and enable it in the mod menu or mod manager of your choice, also a mod that uses no scripts but has a bsa archive will not work out of the box because the SE uses a different archiving system more similar to Fallout 4's, BUT if you extract the archive using a mod manager and then install the extracted files directly into the data folder for SkyrimSE it will work for most mods. There are plenty of downsides for doing it this way and I dont suggest doing it and just waiting for the mod authors to port the mods normally but it IS possible to use certain older mods with the Special Edition.