Simply put this is a rhythm (with Drums, bass and chords) that can be used to perform songs ... etc. A style (e.g. COUNTRY-1) is simply just that ... a style. You can choose to use this style to play many songs ... depending on your repetoir. You can EDIT/Modify a style (e.g change the instrumets the various players (bass, chords, drums) are playing ... etc).
Now, let's say you are on stange, and someone requests ... "I will Always Love you" by Whitney Houston. You will have to 'search' for the appropriate STYLE (e.g. 8 Beat 1), search for the appropriate tempo (e.g. 80), set up the voice you would want to use ... (e.g Sax with a 2nd voice of Strings), set up the Transposer values (if/when applicable) ... etc. From the time someone requested the song to when you actually start playing could take you about 2mins (depending on how familiar you are with the instrument).
HOWEVER .... suppossed you had done all this at home a while back and saved the setting (song setting) into a REGISTER (e.g REGISTRATION 001?) Then ALL you would have to do is press REGISTRATION - key in "001" and instantly all the above settings are recalled for you to start playing the song.
Registrations are simply SONG SETTINGS. A REGISTER controls the style, voice, pedal settings, panel status, Midi settings ... etc of the keyboard at a given time. Depending on the keyboard, you can setup over 190 REGISTRATIONS (or song settings) so that at the show, when you want to go from one song to another, you simply TYPE in the co oresponding registration number - which is a lot easier and faster than trying to search for the style, sounds and other keyboard settings on stage.
Hope this is clearer.