'Tape out' is sending some data to the fab, thereby initiating some part of a manufacturing process for a chip. As Simon says (ha!) the origin of this is from the days before high bandwidth connectivity where the data had to go on tapes.
This might be a whole chip description, or only part of it. For example, it is possible to tape out the base layers a few days/weeks before the metal layers are ready, or just to tape out metal for a respin.
Also note that even 'production' silicon can be respun (yield and bug fixes would be the main reasons for doing so).