Issue 2 of dig has a neat article about what happens to artifacts. If you read the article, you'll find out that that once an artifact like a pot is removed from the soil, it is cleaned and recorded.
Archaeologists mostly find fragments of pottery, called potsherds.
Whole pots, even if they are broken, are pretty rare. So many potsherds are found at a site that not all of them can be put back together again. Only the most important remains can receive such special attention. When important remains of a pot or fragments of a pot are discovered, they are taken to a laboratory to be cleaned and pieced back together again with special glue. This is called conservation.