Lines intersections with a point in 1D
If you store all of the intervals in a segment tree, then the operation "find all intervals that contain the point $x$" is known as a stabbing query.  Building a segment tree from $n$ intervals takes $O(n \lg n)$ time, but once it is built, you can answer a stabbing query in $O(\lg n + k)$ time, where $k$ is the number of intervals that intersect the point $x$ (i.e., the length of the output of the operation).

In comparison, a linear scan (as described in G. Bach's answer) requires no preprocessing but takes $O(n)$ time to answer each stabbing query.  Thus, if you plan to make many stabbing queries on the same set of intervals, it may be faster to build a segment tree in advance, as each stabbing query can then be answered much faster.

See also interval trees, which have similar properties.