A base case is the point in a situation where a player is 100% sure of the correct move without reading at all. Base cases are different for different levels of players.
A player of about 15k should be able to tell that a is the correct move for either player without reading at all. This is a base case for such a player.
Of course it is possible for a player to read out the possibilities in Variation A and Variation B, but beyond a certain level this thinking process is rarely ever completed. A player saves time by remembering the base case.
Base cases don't necessarily mean memorized positions. At the lower levels, it is possible (and perhaps even beneficial) to memorize some base cases (unsettled eyeshapes, net and ladder shapes) but the vast number of possible shapes is prohibitive beyond very basic situations. Instead, base cases are usually determined by intuition.
Take, for example, this situation:
At my current level (3k) I know that white needs no further moves to kill without any reading. If asked to explain to a lower-ranked player I could play out some variations that would demonstrate that a b and c make a sort of miai, but in an actual game I would not even think of this; I would simply know intuitively that black doesn't have enough eyespace and there are too many vital points to protect at once. I also didn't memorize this situation (though I have seen it before in games).
The importance of this to learning go is that one can improve one's reading by expanding the complexity of one's base cases. This leads me to develop the base case problem learning method.