Lighting angles are relative positions of the light fixture, and the target it's illuminating. Different angles can be used to convey a mood or tone in support of the overall production design.
A figure lit with front light
Front Light is generally used for most "utility" Illumination, many lighting designers use it as a starting point for their design. This is because front light generally illuminates everything of importance, actors' faces, set pieces, and props. Because of its angle relative to the audience, it can often make objects in its beam look flat, thanks to its position it tends to hide shadows from the audience, which is one of the important cues that can be used to sense depth.
A figure lit with down light
Down Light can help create a sense of depth by casting large shadows over objects in its beam. A good choice for second pass lighting design, down light can help with things like flooding the stage to create a sense of space, and environment before action specific lights are brought in.
A figure lit with back light
Back Light is one of the least visible angles you can cast, it helps to cast a silhouette around anything in its beam. This effect can be desirable for visually separating an object from its background, or for creating a sense of belonging (or not belonging) in an environment.
A figure lit with left Side Light
Side Light helps to create a more extreme, less subtle outline, of anything in it's beam. It is often used to highlight the physical form of a figure, as opposed to a character on stage. For this reason it is commonly found in lighting for dance. It is however still useful in a purely theatrical context.
A figure lit with up light
Up Light can be used to create a sense of uneasiness on characters in particular. This is in part because this angle is not commonly found in everyday life, and because it can cast unnatural looking shadows on figures, and backgrounds.