Pikachu’s Neutral Special is Thunder Jolt: a simple attack that seems to be inspired from moves like thunder shock. It’s a nice projectile that allows you to pepper the opponent with electric shocks, but it’s ultimately not much to write home about.
The name of the moves comes from the TCG. In fact, this was the attack listed on the first ever Pikachu card, so it holds historic significance.
Side Special is Skull Bash, a move that Pikachu could only learn via TM in Gen 1 and never any other. In the games it was a two turn move like Solar Beam, wherein the user tucks their head in for the first turn and launches the next.
It is one of the most faithful interpretations of a Pokémon move to Smash Bros. it is made into a chargeable move carrying the original intent.
Despite its faithful interpretation, though, I cannot help but find the move a bit misplaced. Not only was this move outdated on Pikachu by the time it was introduced in Melee it also takes the generally speedy move set of Pikachu and brings it to a halt.
Up Special is Quick Attack, a staple move that anyone raising a Pikachu is familiar with. In the games this allows a priority hit on the target playing into Pikachu’s speedy nature.
The same is true in Smash, as it allows Pikachu to zip twice around the air to do damage or make an escape.
Interestingly, Smash adds electrical elements to Pikachu’s quick attack which is a bit of foreshadowing for the move Zippy Zap as seen in Let’s Go Pikachu and Eevee which is essentially an electric quick attack that is exclusive to the partner Pikachu.
Down Special is the mighty Thunder, the most powerful electric type attack in Gen 1. Pikachu stands in place and uses it's tail as a lightning rod (a reference to Pokédex entries) to summon a bolt from the heavens.
This is similar to how the move is depicted in the Pokémon anime and Pokémon Stadium.