What exactly is the meaning of an "ark" that applies to a boat and that thing that melts off the Nazi's faces in the first Indiana Jones movie?
We need to be careful thinking that our English (or other language) translations always reflect God's Words perfectly. The word 'Ark' when referring to Noah's ark or the ark of bullrushes that Moses was placed into is the hebrew word 'teebah'. The word used for the Ark of the Covenant is 'arown'. I believe that there are similarities between the two as noted in other answers, but the words given are distinct in the Hebrew.
An ark isn't a boat - it is a place of refuge - a container that protects things.
Jews place their Torahs in an "ark" - a special box made to preserve the contents.
The ark of the covenant was a box that protected and preserved the 10 commandments, Aarons rod, and an omer of manna.
More importantly, despite the fact that when you say "ark" most people think "boat," the point of the ark was not that it was seaworthy, but rather that it preserved the lives of the people stored in it.