MongoDB: How to find out if an array field contains an element?

It seems like the $in operator would serve your purposes just fine.

You could do something like this (pseudo-query):

if (db.courses.find({"students" : {"$in" : [studentId]}, "course" : courseId }).count() > 0) {
  // student is enrolled in class
}

Alternatively, you could remove the "course" : courseId clause and get back a set of all classes the student is enrolled in.

[edit based on this now being possible in recent versions]

[Updated Answer] You can query the following way to get back the name of class and the student id only if they are already enrolled.

db.student.find({},
 {_id:0, name:1, students:{$elemMatch:{$eq:ObjectId("51780f796ec4051a536015cf")}}})

and you will get back what you expected:

{ "name" : "CS 101", "students" : [ ObjectId("51780f796ec4051a536015cf") ] }
{ "name" : "Literature" }
{ "name" : "Physics", "students" : [ ObjectId("51780f796ec4051a536015cf") ] }

[Original Answer] It's not possible to do what you want to do currently. This is unfortunate because you would be able to do this if the student was stored in the array as an object. In fact, I'm a little surprised you are using just ObjectId() as that will always require you to look up the students if you want to display a list of students enrolled in a particular course (look up list of Id's first then look up names in the students collection - two queries instead of one!)

If you were storing (as an example) an Id and name in the course array like this:

{
        "_id" : ObjectId("51780fb5c9c41825e3e21fc6"),
        "name" : "Physics",
        "students" : [
                {id: ObjectId("51780f796ec4051a536015cf"), name: "John"},
                {id: ObjectId("51780f796ec4051a536015d0"), name: "Sam"}
        ]
}

Your query then would simply be:

db.course.find( { }, 
                { students : 
                    { $elemMatch : 
                       { id : ObjectId("51780f796ec4051a536015d0"), 
                         name : "Sam" 
                       } 
                    } 
                } 
);

If that student was only enrolled in CS 101 you'd get back:

{ "name" : "Literature" }
{ "name" : "Physics" }
{
    "name" : "CS 101",
    "students" : [
        {
            "id" : ObjectId("51780f796ec4051a536015cf"),
            "name" : "John"
        }
    ]
}