As stated, ship is short for relationship, so if you ship a pair then you take pleasure in seeing/imagining them as a couple. This can be "canon" couples (ones depicted as together on screen in the series) - eg Bangel is the portmanteau for Buffy/Angel - or non-canon which are couples that weren't officially together, but fans can enjoy imagining them as such - eg Fuffy (Faith/Buffy).
It's not necessarily true that to ship a couple you have to "love them very much above all other couples" as GrrArgh says. Far from it. Most fans will have a long list of ships they like and will often ship the same character with a couple of different pairings. For example, let's take the character of Faith. First and foremost Fuffy, but then I also like Faillow (Faith/Willow) and when in the right mood, Fawn (Faith/Dawn) and even on to other characters like Wesley, Giles, Anya etc.
Regarding BtVS then you'll find that the most controversy, or arguments, or shall we say loyal fans revolve around the two pairings of Bangel and Spuffy. Those two choose to take a side and will aggressively denounce and put down the other (usually with humour and not too seriously, but some take it very seriously.) There are some that can enjoy both pairings, but it is rarer than with other ships. I think xfivebyfive put it best when she said that the Bangel and Spuffy fans seem to carry over the Angel/Spike fued to their respective ships.