A table lamp is geared to provide an alternate source of lighting for a room, beyond natural light (from windows, etc.) and overhead lighting (light wired into your ceiling).
Table lamps can provide practical pools of light: for example, a desk lamp, or a side-table lamp that casts illumination on a specific person or work-space. They can also be used in concert, two or three to a room, to create a subdued or intimate environment with lower lights.
Ultimately, there are as many uses for table lamps as there are tables, and, just like tables, they are extremely helpful in a number of different scenarios. We'll go more into these below.