How do I group elements in a view? I have 8 articles: Article 1 (Gaming), Article 2 (Local News), Article 3 (World News), Article 4 (World News), Article 5 (Gaming), Article 6, Article 7, Article 8.

In a view I would like to group the articles by: "News," "Gaming," and "Other." Combining the world and local news under one category, and moving anything that doesn't have a category to "Other." I didn't want to create a new vocabulary for just this view. I tried to accomplish this using the group by field, but I can't figure out how I could accomplish this using that method.
In your content type, you can create a "list" field called, say, Category, with the categories News, Gaming and Other. You can then categorize the content that should appear in your View using that field. 

In your View, under Format, choose "Settings." There should be an option to choose a Grouping Field. Select the "Content: Category" field. Your view should now show up grouped according to News, Gaming and Other.

Was that what you had in mind? 

UPDATE
The above wasn't quite what you were looking for. No problem. 

You say you have articles like Article 1 (Gaming), Article 2 (Local News), Article 3 (World News), Article 4 (World News) etc. Is it right that the word in parentheses is a taxonomy term? If not, the below won't be relevant, but I'll assume for now it is.

If so and you want to 'chunk' these terms into three categories, you could certainly modify the Views query or write your own. 

Alternately (which is probably easier) you could change your existing vocabulary into a hierarchical taxonomy with News as the super-category and Local News and World News as the subcategories.

So now your vocabulary looks like:

News


Local News
World News


Gaming

Other

If your content was already tagged with the existing categories, you shouldn't have to change anything.

You could then create a View that displays the fields you want. Add a filter "Term Name," and enter "News." Then, add an attachment display, and attach it to your current View, overriding the filter and setting it to "Gaming." Finally, add a second attachment and override the Filter, changing it to "Other." This should give you the result you want.