Geodesics are locally extremes of length.
In ordinary geometry, paths which take the shortest route between points which are close together on the path. In general relativity, it is sometimes a maximum of elapsed time. Feynman has a cute illustration in a book of his. Suppose you want to arrive back where you are now in one hour of local time, but with a maximum of time having elapsed for you. Note that going uphill takes you to a place where, informally speaking, time goes faster.