One of my players has Talon, a +1 Longsword that deals slashing damage.

In regards to a creature that has "resistance to damage from non-magical weapons", does Talon deal magical or non-magical damage?

In this case: Yes, it's magical

The chest also holds a
+1 longsword in a silver-chased scabbard. The sword is inscribed with the name "Talon" ... (LMoP 24)

All of the properties of the item are given by the description "+1 longsword". This is not a generic description of a bonus, but the designation of a specific item detailed both at the end of the adventure (p.52) and in the DMG (p.213). As such, it is a magic item and bypasses resistance to non-magical attacks (MM errata on MM p.8).

Generally: Ask your DM

If you only know the effects or bonuses of an item, it is best to ask your DM (preferably by casting identify).

In 5e bonuses are not given lightly and weapons that have +X are usually magical items. However your DM is free to invent any item. Possibly the sword is of such craftsmanship that it performs better in combat. In that case it might not be considered a magical item and to overcome resistance to non-magical attacks.

Yes, sort of, but damage works a little differently in this edition of D&D

Magical describes the weapon, and the attack; not the damage

The magical sword makes a magical attack, it doesn't do magical damage. This is a subtle point, but it is worth remembering. (The related case of silvered weapons follows this same rule / logic).

... a magical attack is an attack delivered by a spell, a magic item, or
another magical source (Basic Rules, DM, p. 4; under Vulnerabilities, Resistances, and Immunities).

You describe Talon as a longsword, +1. In the DMG, you will find such weapons under Magic Items, in the category of Weapon, +1¹.

In regards to a creature that has "resistance to damage from non-magical weapons", does Talon deal magical or non-magical damage?

You cited the standard rules block incorrectly.

Damage Resistances bludgeoning, piercing, and slashing from nonmagical
attacks (Magmin; MM)
Damage Immunities bludgeoning, piercing, and slashing from nonmagical
attacks not made with silvered weapons (Werewolf; MM)

Unlike some video games and some other RPGs, D&D 5th edition does not specify magical as a type of damage, although Force as a damage type is described as pure magic that does damage.

Damage Types (Basic Rules, p. 75)

Damage types have no rules of their own, but other rules, such as damage rresistance, rely on the types. The damage types follow, with examples to help a DM assign a damage type to a new effect.

The rules cite the following kinds of damage. (edited for brevity)

Acid. Bludgeoning. Cold. Fire. Force. Lightning. Necrotic. Piercing. Poison. Psychic. Radiant. Slashing. Thunder.

Note that 'magical' isn't one of the listed damage types. Long swords do slashing damage. (Weapons table, PHB).

Weapons typically do Bludgeoning, Slashing, or Piercing damage. Monster stat blocks often show resistance or immunity to Bludgeoning, Slashing, or Piercing damage from non-magical, or non-silvered, weapons. (Werewolf, magmin, as above). Some will also show resistance or immunity to other damage types. A red dragon, for example, is immune to fire damage. (MM; SRD p. 286 also) Fire is a damage type.

How does this work during play?

Ask yourself: is the creature resistant to, or immune from, non-magical attacks?

If yes, then a non-magical weapon does half damage (resistance) or no damage (immunity), but a magical weapon overcomes that resistance or immunity and therefore still does full Bludgeoning (mace), Slashing(sword), or Piercing(spear) damage to the target; it's attack is magical.

What damage does Talon do?

Talon is a magical sword (+1); it makes magical attacks and thus bypasses any resistance or immunity, doing full slashing damage.

Put another way, the sword isn't doing "magical damage" but is rather doing the standard kind of damage because it is making a magical attack.

¹ Under "Magic Items" In the SRD, we find. (DMG ref later)

Weapon, +1, +2, or +3 Weapon (any), uncommon (+1), rare (+2), or
very rare (+3)

You have a bonus to attack and damage rolls made with this magic
weapon. The bonus is determined by the weapon’s rarity.

"Magical Attacks" as a game feature.

A good example of this is the Monk's 6th level ability Ki-Empowered Strikes

Starting at 6th level, your unarmed strikes count as magical for the
purpose of overcoming resistance and immunity to nonmagical attacks
and damage.

The attack is magical, not the damage.

Yes, it's magical

In this case the answer is yes, as Talon is a magical item which confers a bonus of +1 to attack and damage it counts as a magical item for the purposes of negating damage resistance.