Multi-band refers to a raster dataset that has more than one band or channel of information. For example, an RGB image such as a jpg has three bands - one for the values of each color.
In imagery, single band (black and white, luminance, often panchromatic) and three band (rgb, natural color, visible light) files are common. Multispectral imagery often adds a fourth (infrared) band at a minimum and potentially several more at different wavelengths (see Landsat for example). Hyperspectral imagery can have hundreds of bands focused on very narrow wavelength ranges.
Multiple bands can also be present in non-imagery rasters. For example a raster may contain interpolated monthly precipitation readings for an entire year, with each month being a separate band.