The list of changes that have been discovered in Undertale v1.001 have been moved to the changelog article. The remainder of this post will remain here, only for archival purposes.
Toby Fox recently announced beta testing for the bugfix patch for Undertale on Twitter, so here's an announcement of just that. You can only participate if you have the Steam version of Undertale. If you have the Steam version and would like to participate, follow the instructions in this image.
The Steam version of Undertale v1.001 has been updated to fix bugs and increase compatibility. If this causes the game not to work for you, then revert to the old version. The DRM-free builds have now been updated, either find the original download link given to you or use the resender.