Figure skating has four disciplines--Men's, Ladies', Pairs, and Ice Dance. All feature a short program (between 2.5 and 3 minutes) and a long (either 4 or 4.5 minutes depending on the discipline). Additionally, most if not all skaters prepare exhibition programs--non-competitive skating for ice shows, post-competition galas, etc. Exhibitions are often a lot of fun, as skaters can use props and do other things that aren't allowed in competition.
I've linked one example of each kind of program for each discipline, making sure to not repeat any skaters. These are all programs I really love, for a variety of reasons. You don't have to watch all of them but hopefully you check out and enjoy at least a few! I've put a star next to my favourite in each of the four categories.
MEN
SP: Takahiko Kozuka, "Take Five", 2008 Grand Prix Final
FS: Javier Fernandez, "Chaplin Medley", 2013 Eur. Championships
EX: Patrick Chan, I Need a Dollar, 2013 Stars On Ice**
LADIES
SP: Carolina Kostner, "Ave Maria", 2014 World Championships
FS: Akiko Suzuki, "O", 2013 World Team Trophy**
EX: Maé Bérénice Méité, "Partition/Show Me How You Burlesque", Team France Tour
PAIRS
SP: Kirsten Moore-Towers and Dylan Moscovitch, "Micmacs", 2014 Can. National Championships
FS: Ksenia Stolbova and Fedor Klimov, "The Addams Family", 2014 Olympic Games
EX: Meagan Duhamel and Eric Radford, "Ho Hey", 2013 World Championships**
DANCE
SP: Piper Gilles and Paul Poirier, "Mary Popins Medley", 2013 Four Continents Championships
FS: Nathalie Pechalat and Fabian Bourzat, "Le Petite Prince et sa Rose", 2014 World Championships **
EX: Kaitlyn Weaver and Andrew Poje, "Shake it Out", 2012 Cup of China