I found CLRS's explanation of divide and conquer gives a very intuitive understanding of what a logarithmic time algorithm is doing and why it runs in logarithmic time.
A little bit of nitpicking on the accepted answer: the list seems to imply that these are "the" possible runtimes an algorithm may exhibit. But there are many cases where algorithms run in times that lie between these. Perhaps related are log-star and O(log log N) algorithms. These are even more interesting to me than logarithmic ones.