Shipping are fan theories where fans put two characters into romantic relationships - whether or not there is any indication that the characters were attracted to each other in the actual show.
The classic example in Pokemon is "PokeShipping" which is the pairing of Ash and Misty. More recently, "AmourShipping" has become much more popular (Ash + Serena) given the kiss Serena gave Ash. There are a lot of these Shipping theories, though, and some of them are extremely strange, sometimes even involving Pokemon as well as people.
Here is a list of many examples of Pokemon Shipping, if you are interested in specific ones.