Neutral has several different meanings in fighting games at large and Smash specifically.
A neutral input means the control stick is centered, with no directional press registering. This is important all over the place. Neutral specials, neutral attacks on the ground, and neutral air attacks when jumping all require the control stick be centered during the Attack/Special button input. Returning the control stick to neutral during a dash or run allows for non-dash attacks out of the aggressive forward movement (which is a huge change for Super Smash Bros. combat compared to past editions).
The neutral game is the state where all combatants are relatively evenly matched, with no one having a gross frame or position advantage. This is where feinting and poking takes place, attempting to fish for clean hits or bait out overcommitments that can be punished. When opponents joust and spar, but neither is actively being forced to deal with the consequences of a previous exchange, that’s the neutral game.
The neutral game favors quick, low-commitment fishing attacks, like shooting projectiles, tossing items, poking with the tips of neutral attacks and quicker tilt attacks, attempting grabs and dashing grabs and dash attacks that cross up against shielding enemies, and jumping/short-hopping airs with long active periods (fast-falling to return to earth quickly and cover more airspace, naturally).
Laggy, high-commitment moves are risky in the neutral game. This includes more ambitious special moves, most smash attacks, and dash attacks that can’t push through defenders on block. Unless you’re making a hard read based on opponents’ habits, save these attacks for situations where you have a heavy advantage over someone, so it’s less likely they can see it coming and jump, block, or dodge. Examples of situations with a heavy advantage, where you can leverage higher-risk, higher-reward attacks (like charged smashes or laggy specials), include when you’ve just thrown or launched someone, so they have to tumble and perhaps tech on the ground, or when you punt someone clear off the stage, forcing them to focus on recovery instead of combat.