A covering index is an index that contains all of, and possibly more, the columns you need for your query.
For instance, this:
will typically use indexes to speed up the resolution of which rows to retrieve using criteria, but then it will go to the full table to retrieve the rows.
However, if the index contained the columns column1, column2 and column3, then this sql:
SELECT column1, column2
and, provided that particular index could be used to speed up the resolution of which rows to retrieve, the index already contains the values of the columns you're interested in, so it won't have to go to the table to retrieve the rows, but can produce the results directly from the index.
This can also be used if you see that a typical query uses 1-2 columns to resolve which rows, and then typically adds another 1-2 columns, it could be beneficial to append those extra columns (if they're the same all over) to the index, so that the query processor can get everything from the index itself