A399
Unarmed attacks are weapon attacks, because they're not spell attacks and those are the only two highest-level categories of attacks[1]. However, they are not attacks with weapons, which matters for some abilities (e.g. Improved Divine Smite).
Thus, a Samurai who uses Fighting Spirit and then punches someone has advantage on the attack roll.
[1] The hierarchy goes:
1. Weapon vs Spell. This determines what modifier is applied (in the absence of other features)--Weapon attacks use STR or DEX, depending on the weapon. Spell attacks use INT/WIS/CHA depending on the feature that granted them. Spell attacks are also automatically magical.
2. Ranged vs Melee. Ranged attacks incur disadvantage if an enemy is within 5 feet, even if that enemy is not the target of the attack. Melee attacks do not. You can make a ranged weapon attack with a melee weapon by throwing it (either via the thrown property or as an improvised weapon or some other feature). You can make a Melee Weapon Attack with a ranged weapon by using it as an improvised weapon.
3. With a(n) [melee|ranged] weapon/-- Some abilities specify that they only apply to attacks made with a specific type of weapon (usually melee or ranged). These exclude Unarmed Attacks, because Unarmed Attacks are not weapons, despite being used for Weapon Attacks. Some are even more specific--Melee Weapon Attack with a Melee Weapon is a theoretically-possible (although I'm unaware of any that are that specific).
The statement is always [melee|ranged] [weapon|spell] attack [with a [qualifier] weapon]. Things in [square brackets] are not necessary--if not stated, they apply to all categories of that value. Thus, if it just says "you have advantage on attack rolls", that means all attack rolls, regardless of source. Fighting Spirit specifies weapon attack rolls, so it doesn't apply to spell attacks. But applies to all non-spell attack rolls.