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What is the structure that SQL Server uses to build and maintain indexes called?

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They are generally B+ Trees for your standard clustered and non-clustered indexes:

B+ Tree Indexes

Columnstore indexes have a different structure:

Columnstore Indexes

In SQL server 2014 you have some new types too for the in-memory tables such as the Bw-Tree index:

Bw-Tree Indexes

And the hash index:

Guidelines for Using Indexes on Memory-Optimized Tables

Not forgetting the full text indexes too:

Full Text Indexes

As pointed out there are also spatial indexes:

Spatial Indexes

And XML Indexes:

XML indexes

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There are also the spatial indexes. –  ypercube Nov 13 '13 at 13:59
    
Good point, and XML indexes, which I'd forgotten about too –  steoleary Nov 13 '13 at 14:00
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