What exactly is a class in kNN search?
Given the context of your question I assume your are referring to the k-NN classifier. The idea of the classification method is you have a set of feature vectors $f_i$, in general, $f_i \in \mathbb{R}^d$  where $d$ is the dimension of the vectors. Additionally you have a class $c_i$ for each $f_i$. To classify an unseen feature vector $f_j$ you select the $k$ nearest neighbors (under some distance, usually the euclidean distance) and the most common class among the neighbors is your prediction. 

So, summarizing, the input of the algorithm are the feature vectors $f_i$ and the classes $c_i$. For more on the algorithm see this wikipedia link. 

The image (from wikipedia) below illustrates a typical example, in this example the classes are: red triangle,blue square and the point to predict is the green circle. Each feature vector is a two dimensional point, i.e. $f_i, f_j \in \mathbb{R}^2$. In this situation the predicted class will be red triangle.



For an explanation using both Python and R see this.