'Shipping' is the belief that two (or sometimes more) characters of the show will, are already in, or -in your opinion- should develop a relationship with one another. Hence the name 'shipping. People come up with names for them, ending in -shipping. Most don't make much sence, but they usually describe something about the couple. For example, Orangeshipping, which is Tracey ♥ Misty, is named because they met in the Orange Islands. Others, such as Conceitshipping(Gary ♥ Lily), are named because of the personality they share. Many also have no very obvious meaning behind them, like Summonshipping, Falkner ♥ Melody.
A few even have characters who are related in couples together, like Blackthornshipping (Clair & Lance, who are cousins) and Petalburgshipping (May & Max, brother/sister).
Pokemon can be in shippings too, either together, or with a human.
That's basically it. I'm tired of typing. xD