I have a simple search form set to look in entries. When I search for the category name that I have selected in some entries, the entry does not show in the results. Should I be able to use Simple Search to look in an entries selected categories?
The EE search module searches the content of entries - it will not look for names of categories. You may need to look at some of the third-party options for that.
Here's a quick list of add-ons that that may help out. Look to Low Search and Solspace Super Search first, would be my suggestions. See this search for 'search' on Devot-ee