Feebas's method of evolution is somewhat unique as it has more than one way to evolve which will largely depend on the version you're playing. It is one of three Pokémon––the other two being Espeon and Umbreon––who've had their evolution methods changed (the others though was only due to the lack of an internal clock within Pokémon XD). Here are the different ways to evolve a Feebas into a Milotic.
Making its appearance in Pokémon Ruby/Sapphire was also the introduction of Pokémon contests and alternate methods of evolution. In these games you could evolve Feebas by raising its beauty level to 170 by feeding it blue Pokéblocks made at contest halls by using a Berry Blender. When doing this it is best to have a Pokémon with a nature that likes blue Pokéblocks––Modest, Mild, Quiet, and Rash are the best natures to go with as they like the flavor and will benefit the most from them. Also, because there are no Pokémon contest halls in Pokémon FireRed/LeafGreen, the only way to evolve a Feebas was to trade it to Ruby/Sapphire/Emerald and raising its beauty up in those games.
From Pokémon Diamon/Pearl to present Feebas can be evolved by a less exciting, but much easier, means involving trading it while holding a Prism Scale. However, in Pokémon HeartGold/SoulSilver it is possible to evolve Feebas into Milotic without trading. This can be done by visiting the Haircut Brothers in Goldenrod City or by visiting Daisy in Pallet Town to groom your Pokémon. In addition to raising a Pokémon's friendship it can also raise a Pokémon's beauty, a stat which, although invisible, still exists within the structure of each individual Pokémon. However, this method will not work if its sheen is too high. It is definitely something worth trying out if you don't have a Prism Scale.
In Pokémon Omega Ruby/Alpha Sapphire Pokémon contests make a return and Feebas's old method of evolution is back! Getting the appropriate Pokéblocks is also much faster and easier now as the Berry Blender has been retired from the contest halls and is replaced with a Pokéblock Kit, an item found in your Key Items tab which you can use whenever and wherever you want. It's awesome.