In addition to many other drawbacks wordpress search has it just can’t search the description associated with a taxonomy (category, tag) or author, so even if the most obvious search search result is the category page, the internal search will never show it.
But there is a way to hack around it if you really have to. All that needs to be done is to have a page with the exact same URL as a taxonomy.
If you which for the category “events” to be searchable, assuming its url is /category/events, all you have to do is to create two pages, one with the slug “category” and a sub page of it with the slug “events” and put the text associated with the category in the “events” page.
The only problem is that the search result will be styled like a page, but this is a small price to pay.