According to the Urban Dictionary, "it generally refers to romantic relationships, but it can refer to platonic ones as well". The Urban Dictionary also mentions that the phrase "ship" comes from the term "relationship".
When a couple is shipped, that usually means that their names are mushed together. For example, Percy Jackson and Annabeth Chase of Rick Riordan's Percy Jackson and the Olympians have the shipping name Percabeth. Or, in other cases, two titles can be meshed: Captain Hook and Emma Swan of ABC's Once Upon a Time have won the shipping name Captain Swan.
Where am I going with this? Some authors experience a little issue when they're writing about a romance between two people who are just not meant to be together -- they become attached. Even though it might not be in the characters' destinies to be wed and grow old together, the writer will still notice the flaws in the relationship -- and just how much the two characters are meant to be together.
I like to call this problem one of the many "writer's issues". Seeing as the writer often falls for the characters even harder than the readers do, they find destroying the two characters' relationship incredibly heartbreaking. And they're stuck.
What to do! Some writers shove aside their feelings and ruin the characters' romance; others simply sigh and look for a way around it. My advice? Take serious note of all the flaws in the relationship, and move past it. Remember that the whole point of their failing love is for your story. Just don't do anything rash...
Image author unknown; recovered from this link.