At the reading settings page, it says you can pick whether articles show the full text, or just the summary.
I clicked on the summary, but my posts show up as full text. Why is this? Is it because of the theme I have?
No, your posts are showing up as full posts because full posts are the default. What that setting changes is only how the posts show up in your RSS feed. If you want your actual posts to be truncated on the front page, you must either use a theme that does that automatically, or you must use the Read More tag: