Depth of Field is the perceived sharpness of focus across multiple regions of a photo. The photo to the right has a narrow depth of field in that only a small portion of the overall picture (the budding tree stem) appears in focus while the background is blurred. This relative sharpness makes the viewer focus on the subject while providing context and fill to the rest of the frame. Cognitively the viewer assumes that the background is farther (deeper) in the distance giving a perception of a third dimension.
If you're still shooting in Auto Mode, then the camera is making all the choices for you including your depth of field based upon your focal point and the amount of available light. Take control of depth of field using the aperture setting. Aperture refers to the area of the opening for light to pass through the lens. A lower aperture setting (measured in F-stops) like F2.8 allows more light and creates a narrow depth of field. A high aperture setting of F11 will reduce the depth field bringing similar degrees of sharpness to the entire frame.
A narrow depth of field is idea for portraits when you want the viewer to focus on the subject without the distractions that the background provides. Floral shots (like the one above) also benefit from a narrow depth of field, because the complexity of flower is sufficient and a sharper background could overwhelm the photo. Conversely, landscapes and cityscapes benefit from a broad depth of field where the majority of the frame is in focus giving a sharpness to the broader subject. Sports photos benefit from both broad and narrow depth of field. Sometimes the desired effect is to focus on the player while in other instances the intent is to focus on the broader play and the relative positions of the players.
Depth of Field is an important ability that most casual photographers have difficulty controlling. However, it is one of the most critical concepts in photography and once mastered will dramatically improve the quality of your photos.