To be technical:
1. SMB3 uses a feature of the NES that "clips" the left 8 pixel columns of the video display. These pixels are drawn using the current screen background color (in general, the color of the sky in levels that take place outside). This is done to allow smooth sprite scrolling off the left side of the "playfield" portion of the screen, as well as to minimize the visual artifacts caused by the 2D scrolling (see below).
2. SMB3 supports two-dimensional scrolling, even though there isn't enough VRAM to support it "natively." As a result, the contents of the screen that scroll off the right edge will "bleed" into the left side of the screen and vice-versa. Although the bleeding is mostly invisible because of the 8-pixel clip mentioned above, the effect is still visible in the color scheme used to draw the background graphics (each entry in the NES's color table works on a 16x16 region of the screen, which is wider than the 8-pixel margin). Note that many games that scroll in both directions choose to map the VRAM differently, causing the artifacts to appear at the top and bottom of the screen instead (which is much less likely to appear on a TV, due to the NES's resolution being "taller" than the TV's actual dimensions).
3. To display the status bar at the bottom of the screen, the game uses a special "raster" effect to change the portion of VRAM to be displayed (as well as which character ROM banks should be used as the graphics source). The game has to disable video output temporarily to do this (which is why there's a 1-pixel horizontal line between the playfield and the status bar). However, due to timing issues (especially when certain sounds are playing), SMB3 is unable to turn on the display in time for the first few pixels of the status bar to display properly.
Other glitches show up in other parts of the game, such as the Spade game (not the N-Spade one). Look closely at the left/right edges and you might notice them as you play.