A shipper refers to someone committed to a particular relationship on a show, often seeing this as the most important part of the narrative.
There are two manifestations of shipping. The first is canonical and refers to a relationship that exists in the text eg John and Aeryn on Farscape, Willow and Tara on "Buffy". Often the relationship may be one that is in a state of "UST" or unresolved sexual tension for most of the time eg Mulder and Scully on the "X-Files". Fannish tensions arise when one party in the relationship is attracted to more than one character eg : Kate with both Sawyer and Jack on "LOST".
The second manifestation is when shipping is based around fanon pairings - that is sexual relationships that don't exist in canon, but in fanfiction. This is the case in Superantural, where neither Sam or Dean have ongoing romantic relationships on the series. In fiction however the two main groups of shippers are those supporting Wincest and those supporting Dean/Castiel.