What does it mean when a recipe calls for coring a tomato?

The coring of tomatoes has a lot more to do with tradition than any sound foundations; the seeds in tomatoes can be bitter (and of course will ruin the texture of a smooth gazpacho) but the pulp surrounding them that is also discarded is rich in umami containing compounds:

http://www.guardian.co.uk/lifeandstyle/wordofmouth/2007/oct/23/hestonblumenthalstomatosauc

It proves something many cooks already knew or suspected - that the jelly around the pips of tomatoes contains most of the tomato's flavour. This is not so important for us home cooks - we tend to leave the seeds in the final dish we are preparing; but in the professional kitchen, the seeds are often discarded for the sake of appearance and presentation.

The title of Heston's paper that recently appeared in the Journal of Agricultural and Food Chemistry is Differences in Glutamic Acid and 5'- Ribonucleotide Contents between Flesh and Pulp of Tomatoes and the Relationship with Umami Taste (with co-authors Maria-Jose Oruna-Concha, Lisa Methven, Christopher Young and Donald S. Mottram from the University of Reading).

The paper's abstract cites the difference in taste we've all observed "between the outer flesh and the inner pulp of tomatoes," and the fact that "the pulp, which contains the seeds, had more umami taste." Umami, discovered by Ikeda in 1909, is the fifth taste, the meaty, broth-like or savoury taste that is now accepted as an addition to the traditional sweet, bitter, salty and sour gustatory sensations.

The paper shows that the inner pulp of the tomato contains up to 11 times the concentration of compounds associated with the umami taste as does the other flesh. This was determined experimentally, subjecting many different varieties of tomato to both chemical analysis and to sensory evaluation by a panel of human beings, who wore noseclips "to evaluate taste attributes" and took them off "to assess all other attributes." And just to make sure they weren't prejudiced by visual clues (such as the deeper red of some tomatoes) all the tests were done under red lighting.

Especially regarding tomatoes, there can be some discrepancies on what coring means. In some cases, they mean to just scoop out the stem and the tough white bit under it, and in others they mean to remove the whole central bit with seeds. In my experience "cored" usually means just removing the stem and white bit, whereas "cored and seeded" means removing the stem along with the seeds and central part. This is just a rule of thumb, as recipe authors may vary on their usage.

"Coring" means that yes, you should remove the whole central part of the tomato - the seeds as well as the lighter, slightly fibery part to which they are attached. You only use the external "walls" of the tomato, which have a much more uniform texture and taste.

It is not imperative to core tomatoes. I almost never do it. I eat the pulp, seeds and skin of raw tomatoes (e.g. in a salad) as well as tomatoes intended for cooking. This is what you saw in the later videos. Even when the whole tomato is used, people usually remove the green part just under the stem, because it is very tough and has a strong taste. However, this is not called "coring", and results in a different texture than a cored tomato.

If you don't want to throw away half the tomato, you can just choose a recipe which uses whole tomatoes as opposed to cored ones. I wouldn't recommend to make substitutions if you are new to cooking, this is more of an intermediate skill. Instead of just trying your current recipe with whole tomatoes, look around for another one which doesn't specify coring, there are lots of them.

and then there are those who view "Coring' as removing the stem and green/white part under the stem... http://www.myrecipes.com/how-to/video/coring-tomato-10000001847232/