What does the k-value stand for in a KNN model? What is the k-value in a KNN classification model? Is K the number of Clusters?
 A: The $k$-nearest neighbours algorithm uses a very simple approach to perform classification. When tested with a new example, it looks through the training data and finds the $k$ training examples that are closest to the new example. It then assigns the most common class label (among those $k$ training examples) to the test example. 
$k$ is therefore just the number of neighbors "voting" on the test example's class.
If $k=1$, then test examples are given the same label as the closest example in the training set. If $k=3$, the labels of the three closest classes are checked and the most common (i.e., occuring at least twice) label is assigned, and so on for larger $k$s.
When you build a $k$-nearest neighbor classifier, you choose the value of $k$. You  might have a specific value of $k$ in mind, or you could divide up your data and use something like cross-validation to test several values of $k$ in order to determine which works best for your data. For $n=1000$ cases, I would bet that the optimal $k$ is somewhere between 1 and 19, but you'd really have to try it to be sure. 
A: Knn does not use clusters per se, as opposed to k-means sorting. 
Knn is a classification algorithm that classifies cases by copying the already-known classification of the k nearest neighbors, i.e. the k number of cases that are considered to be "nearest" when you convert the cases as points in a euclidean space.
K-means is a clustering algorithm that splits a dataset as to minimize the euclidean distance between each point and a central measure of its cluster.
Typically, Knn works this way:


*

*You'll need a training set with cases that have already been categorized. The cases will become points in a euclidean space, so you'll want the values to have some kind of normalization so that they are on the same scale.

*Take the first case in the data you want to categorize. Calculate the distance (usually, euclidean distance) between this case and every cases in the training set.

*Select the k training cases that have the smallest distance and look at their classification. These are the k Nearest Neighbors, or kNN. According to the "if it quacks like a duck and walks like a duck it must be a duck" principle, if a majority of it's kNNs are classified as X, then the test case must also be a class X. So you just take the most popular classification of the k nearest points and you're done.

*Do that for every other cases in your dataset.


So k could be described as a parameter that determines the "depth" of the algorithm (how many points will be considered in the classification). As suggested by Nadeem Inayat, a rule of thumb is to use the square root of the training set sample size as a starting point. However, the optimal value for k depends on the context: are you trying to minimize the total number of misclassified points? Or are some  misclassifications (e.g. false negative) considered to be worse than others?
