Much like any other sprite, the player (Mario or Luigi) requires different graphics to be shown, so that vivid animation can be emulated. SMW does this by making a couple of RAM addresses point to various graphic sets for the player, or for attributes of the player.
This address controls which image must be shown of the cape. This obviously only has effect when the player has a cape at all. A diagram of it is shown below.
While walking and sprint-jumping, frames 3 through 6 are cycled through, while 7 through A are used when falling downward. Pose zero is used for both standing still and jumping straight upwards, but is never shown without being preceded by frames 2 and 1.
Note that when the player is swooping/flying with his cape, $7E:13E0 will override the cape image. See also the diagram shown under $7E:13E0, poses #$2A through #$2F. Whatever value $7E:13DF has will be irrelevant during this phase.
Cape images beyond #$0A are not recommended for usage, as they are glitched.
This address controls the "main" poses of Mario and Luigi. Adjusting $7E:13E0 will adjust the player's image as such. A diagram of it is shown below in this paragraph.
Of note are the following poses. Poses #$40 and #$41 are blank - the player is "invisible". These are indicated with a large red cross. Poses #$2A through #$2F absolutely require the player to have a cape powerup, otherwise they aren't shown correctly. Pose #$42 (inflated, small) can be shown regardless of powerup. It is obviously not recommended to be used as such. The same can be said about pose #$43 (inflated, big). Poses #$00, #$3D and #$46 are all used in the powerup animation. Poses #$14, #$20, #$21, #$27, #$28 and #$29 are all used during Yoshi riding. Any pose beyond #$46 is not recommended to use, as they can be glitchy.
Additionally, Schwa made a document that lists the player's poses. It can be found here. Not all descriptions appear to be very descriptive about where or when the pose is used, though.