Figure and Ground
The concept of figure and ground can seem obvious in an artwork. Look! There's a figure and the one color painted everywhere else is ground. But sometimes it's not so obvious and depending on how you view an artwork, the figure can also be seen as the ground, then the figure again. This is demonstrated well with the artwork of the two people kissing, or is it actually a vase in the middle? A more simple example is shown below. The hacker or figure is blue, while parts of him like his collar and his laptop are part of the ground.
Figure and ground is important because figures make a piece of art interesting, and gives the viewer something to look at. Meanwhile the ground provides a rest for the eyes. If a work is too busy then it can actually hurt to look at, which is not what you want.
Frame and Surface
The frame and the surface of an artwork is chosen before a work even begins. For example, in photography people are often told to not put the main object directly in the middle, because if it is off to the side it creates a more interesting photo. There are two "rules" that artists use to guide framing their works. One is the rule of thirds, in which an artwork's beginning canvas is split into 9 equal sized squares. On the four main points is where the viewer's eye will drift, so an artist can put a major viewpoint on those points to make a pleasing picture. The second "rule" is actually the naturally occurring fibonacci spiral. The spiral decreases in size until the end point of the spiral rests on the picture, along with the viewer's attention.
Notice how in this picture your eye is drawn to the elephant? And that cropping it (changing the frame) changes the picture?
Pattern and Rhythm
There are two things to clarify here. Pattern has repeated elements while rhythm has some variation. Patterns are found in thousands of artworks. From old wallpaper to modern day sofa pillows, patterns are everywhere. Patterns and rhythm can make an artwork more interesting. Instead of plain colors patterns give our eyes something to look at and follow around a picture.
Texture gives the viewer a feel of the picture that stimulates physical touch rather than an emotional response. Texture can be real, like in 3D artworks that you can touch but shouldn't. Or Texture can be implied in 2D works like fur or the smoothness of a stone. Texture stimulates the physical sense of touch which can be done through layers and layers of paint (so much so that it is 2D) or it can be just implied flat lines that look incredibly realistic. Either way, texture adds to the realism or just pleasantness of an artwork.
Like in the example above, a textured artwork allows you to remember and imagine what the object would feel like (the feel of dirty wood in this case).
Transparency and Opacity
I know what you are thinking, artwork can be revolved around transparency? Yes. Yes it can. Photoshop in particular is extremely useful because of each layer's transparency and opacity. If you simply stacked pictures on top of another you wouldn't be able to see anything but the top one. But with transparency it can give the viewer a feel of looking through glass or mist. Transparency can also be used to mess with your eyes and show the background behind a figure through a naturally opaque figure.