In a general sense, the term extinct means no longer active or living. It is sometimes applied to kinds of organisms, to cultures, to religions, to geological processes (e.g. "extinct volcano"), to social movements, and, essentially, to any other object or process to which the concept of life or activity is applicable. It could be argued that it can occur before that, say, if there are only males or females left in a sexually reproducing species, or if the remaining pairs are geographically isolated preventing mating.
In the specific biological/paleontological sense, "extinct" refers to the loss of all living representatives of a taxon (that is, of all the members of a species, a genus, a phylum, etc.).
The actions of humans are often to blame for organisms that become have become extinct in the last few hundred years. Conservation of a species is often needed to stem the animal trade market and logging industries to prevent the extinction of endangered organisms.
Cloning is a new technology that is being developed that could possibly bring back organisms into existence. The woolly mammoth and other organisms whose DNA is preserved such as the Thylacine of Australia are realistic options to be brought back using the cloning technique.
Evolutionists believe that over the history of the earth, many more species have become extinct than have existed at any one time. Sometimes a species is replaced by descendants that have altered their form enough not to be considered the same species anymore. An example of this is the history of the modern horse, which has descended through a series of distinct, and now extinct, precursor species.