I find this to be, by far, the best image in this thread.

I like it because the subject "pushes" out of the right side. Although the subject is prettymuch in the center, the rest of the branch is behind it, out of focus. Interesting!
Here are some tips (I have no idea how much you know about photography so I'm starting from square one, don't be insulted).
Rule of thirds. Imagine a 3 by 3 grid of squares. Now imagine the four corners of the middle square. These points are where the subject should go for an interesting photo.*
(pretend this diagram is good, lolz)
You may find, after choosing a subject, that you can't "make it pop" from the frame. How do you make a subject the subject?
Depth of field. I would define depth of field as the depth, going straight out, that you are focusing on. Check out these bad** examples to show you what I mean:
This one has a bigger depth of field. More stuff is in focus. wtf is the subject?

This one has a shallow depth of field. Ah, the subject is the vase!
The difference between these is the aperture. A higher aperture value (less open) means a bigger depth of field, and a smaller aperture (more open) means a smaller depth of field. You can't use this trick on landscapes, but when you're shooting close up things like shit on a table or a portrait, a shallow depth of field is the way to go.
Also try shooting from funky angles, like from the ground or directly up the side of a building...
So yeah, try shiznit out, and remember, any tips you ever receive are guidlines. Have fun and break rules :).
*Not always the case. As I said, guidlines!
**I not only think these photos are really boring, but they are not great examples of depth of field, as both pictures are pretty shallow... damn my perfectionist tendencies!