What would I use them for? I know map package comprises all the layers but how is a map package different from the others?
A Layer File (.lyr) is a file that stores the path to a source dataset and other layer properties, including symbology.
A Layer Package encapsulates the data, cartography, and other properties of the layer as it's authored in ArcGIS into one easily shareable package
A Map Package contains a map document (.mxd) and the data referenced by the layers it contains, packaged into one convenient, portable file. Map packages can be used for easy sharing of maps between colleagues in a work group, across departments in an organization, or with any other ArcGIS users via ArcGIS Online.
(Quotes all from Esri)
So to put in a few words, a Layer File holds path, symbology and layer settings (but no data). Layer Package holds the layer settings plus the data in one package. A Map Package holds the entire MXD/map information, multiple layers, and the data in one package.