Getting a "not defined" error in my JavaScript code when I think it is defined

 politician.countTotalVotes = ...

That won't work as politician only exists while you create luke or oscar (it then points to one of them). Instead you could do:

luke.countTotalVotes = oscar.countTotalVotes = function() { /*...* };

But for more politicians that gets a bit complicated. You could however just make a function that you pass a politician into:

function countTotalVotes(politician) {
  //...
}

countTotalVotes(luke);

Or you use the power of inheritance.

As I suggested in comments, you can rewrite Politician as a class, which basically gives you a template for reusable functionality. If you're not familiar with the concept, a good google term would be "prototypal inheritance" for follow-up on the subject in general. For now, here's a rewrite of your script using a class:

class Politician {
  constructor (name) {
    this.name = name;
    this.electionResults = [];
  }

  // no need to even write a .countTotalVotes() method
  // just define .totalVotes as a getter that does the work
  get totalVotes () {
    var total = 0;

    for (var i = 0; i < this.electionResults.length; i++) {
      total += this.electionResults[i];
    }

    return total;
  }
}

var oscar = new Politician("Oscar Jiminez");
var luke = new Politician("Luke Spencer");

oscar.electionResults = [5, 1, 7, 2, 33, 6, 4, 2, 1, 14, 8, 3, 1, 11, 11, 0, 5, 3, 3, 3, 7, 4, 8, 9, 3, 7, 2, 2, 4, 2, 8, 3, 15, 15, 2, 12, 0, 4, 13, 1, 3, 2, 8, 21, 3, 2, 11, 1, 3, 7, 2];

luke.electionResults = [4, 2, 4, 4, 22, 3, 3, 1, 2, 15, 8, 1, 3, 9, 0, 6, 1, 5, 5, 1, 3, 7, 8, 1, 3, 3, 1, 3, 2, 2, 6, 2, 14, 0, 1, 6, 7, 3, 7, 3, 6, 1, 3, 17, 3, 1, 2, 11, 2, 3, 1];

oscar.electionResults[9] = 1;
luke.electionResults[9] = 28;

oscar.electionResults[4] = 17;
luke.electionResults[4] = 38;

oscar.electionResults[43] = 11;
luke.electionResults[43] = 27;

// console.log(oscar.electionResults);
// console.log(luke.electionResults);

// these are no longer required
// oscar.countTotalVotes();
// luke.countTotalVotes();

console.log(oscar.name, 'has', oscar.totalVotes, 'votes');
console.log(luke.name, 'has', luke.totalVotes, 'votes');

Move your countTotalVotes function into your create function:

var createPolitician = function (name)
{

  var politician = {}; 
  politician.name = name;
  // set to an empty array so this is defined
  politician.electionResults = []; 
  politician.totalVotes = 0;
  // here politician exists, add a function to the object that you can call later
  politician.countTotalVotes = function() 
  {
     this.totalVotes = 0;
     for (var i = 0; i < this.electionResults.length; i++)
     {
        this.totalVotes = this.totalVotes + this.electionResults[i];
      }
  }

  return politician;

};

The error message should be accompanied with a line and column number. In this case, the error is occurring on line 32:

politician.countTotalVotes = function() 

In the definition of the createPolitician function, you declared a variable named politician. var creates a variable that lives within the function it was declared in. If it is not inside of a function, it creates a global variable.

You left the function the variable was declared in, and thus it isn't usable anymore.

If you think about it, it'd be very ambiguous if you could re-use that politician variable later on. What would it refer to?

If you want to be able to call countTotalVotes() on the politicians, then Patrick's comment about using a Class is a good approach to take. You could also assign the function to the politician object that you return from createPolitician - this would give each instance of the politician its own copy of the function.

It's all about variable scope. Declare the politician object variable outside of the createPolitician function for you to access out of the function. It cannot be accessed out of its scope that is why you get an undefined error.