The answers already provided are very good and informative, so I just wish to add something concerning the "metamorphosis" of the very notion of space of which Grothendieck speaks in Semailles.
Every space has its associated topos, but there are topoi which are NOT spatial. You can define categorically the notion of point of a topos, and this definition corresponds to the usual notion of points when one restricts to spatial topoi.
Now, the fact that there are plenty of topoi with no points basically means that one can do topology in a pointless world: you can still formally define notions of compactness, coverings, and well as most of the standard topological (and even homotopical) machinery, directly in a given topos, regardless of its having points or not.
As it turns out, the passage from point-set to pointless topology is not just an idle game: for instance in physics at the Planck level you may still want to talk of topological and geometric properties of space-time, and yet you have no well-defined points.
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