Characters like Pikachu, Peach, and especially Fox were downgraded in power because they were considered some of the stronger characters to use competitively. This is one way to balance out the characters.
On the flip side, characters considered weak in SSBM were made a lot better. The best example is Bowser: He's now faster, both in running and in attacking, and he hits harder. Same goes for Mr. Game & Watch and Young Link (reincarnated as Toon Link). In addition, the Final Smashes for all three are some of the more powerful ones.
By the way, I apologize if I mentioned any spoilers people didn't want to know just yet.