In this context, a feature is a point of interest on the image. In order to compare features, you "describe" them using a feature detector. Each feature is then associated to a descriptor. When you match features, you actually match their descriptors.
A descriptor is a multidimensional vector. It can be real-valued (e.g. SIFT) or binary (e.g. BRIEF).
A matching is a pair of descriptors, one from each image, which are the most similar among all of the descriptors. And of course, to find the descriptor in image B that is the most similar to a descriptor in image A, you need a measure of this similarity.
There are multiple ways to compute a "score of similarity" between two vectors. For real-valued descriptors, the Euclidean distance is often used, when the Hamming distance is common for binary descriptors.
As a conclusion, we can now understand the distance attribute: it is the score of similarity between the two descriptors of a match.
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