A 64k intro is a demo (a non-interactive multimedia presentation) where the size of the executable file is limited to 65,536 bytes. At demo parties there is a category for this kind of demo. 64k intros generally apply many techniques to be able to fit in the given size, usually including procedural generation, sound synthesis, and executable compression.
The size of 64 kilobytes is a traditional limit which was inherited from the maximum size of a COM file. An intro originally referred to an endless demo where all the action happened on a single graphical screen, often to promote a BBS or a game crack. Nowadays it can refer to any demo written within a strict size limit, such as 4 kB or 64 kB.