For some reason, I cannot get the border around each cell in the table to disappear. The square around each cell always shows up. Any help on this?

Semantically a table should be used if the data in it has any kind of relation. Your table with picture does not have any relation. You should do this with floating div´s.
Anyways the layout of all html elements is defined in a css-file. To find out which definitions apply to an element I use firefox + firebug extension. Firebug shows you all css definitions for an extension + the whole way of inheritence/overloading and the link to the css file.