Tomato paste is made from fresh tomatoes that have been cooked down for several hours and then strained. It is a thick, rich, deep red concentrate that is used for thickening food, particularly stews.
Tomato paste in small cans, which is what I grew up with, is inferior, in my opinion, to Italian tomato paste in tubes, for like much tinned food, it has a harsh metallic taste. In addition, the cans contain too much tomato paste for me to use up at once - I generally add but a tablespoon or two to a dish - and the remainder must then be refrigerated for a few days, frozen, or discarded. The tubed variety has the advantage of keeping well in the fridge for months.