I made a background in Photoshop that I'd like to incorporate into my website (see screenshot). It's 1024 x 768.

What if someone has a larger screen? What am I missing? What should I do differently?

You really can't design something for a specific screen size and expect it to work on other screens.

Your image would work a lot better if the critical part were in the upper left corner, rather than the upper right corner. Then, you could just put some spillover in for the largest screens.

Without any change, your image will display in the upper left corner of a screen with higher resolution.

Depending on the browser, the image will either fail to render or will be rendered with scroll bars on a screen with smaller resolution.

You can choose how it will tile with CSS, see http://www.w3schools.com/css/pr_background-repeat.asp

You can restrict it to tiling horizontally, or vertically or turn use "no-repeat"

hope that helps :)

Should I make it smaller and tile it vertically and horizontally? Or, leave it's size and tile both ways?

smaller flipped and mirrored, and tiled each way

especially if you have shares in an optician