Actually, there is a command to move a collection from one database to another. It's just not called "move" or "copy".
To copy a collection, you can clone it on the same database, then move the cloned collection.
To clone:
> use db1
switched to db db1
> db.source_collection.find().forEach(
function(x){
db.collection_copy.insert(x)
}
);
To move:
> use admin
switched to db admin
> db.runCommand(
{
renameCollection: 'db1.source_collection',
to : 'db2.target_collection'
}
);
The other answers are better for copying the collection, but this is especially useful if you're looking to move it.