There are no such things in D&D 5e, at least not in a way that would be familiar from experience with previous editions.
Special Traits are explained on the following page in the Monster Manual. Special Traits are, well, special: they are abilities designed specifically for that creature and are defined per-creature in the creature's entry.
Immunities and damage resistance special traits tend to be a straightforward labelled list — for example, the Banshee (p. 23) is immune to a variety of damage types and conditions, and is resistant to another list of damage types. This works similarly to previous editions and won't require an adjustment in how you think about it.
Magic resistance, on the other hand, uses the special trait space to define how it works for that specific creature. An example is the Death Knight (p. 47), which has Magic Resistance — it says how it works for Death Knights as part of the creature:
The death knight has advantage on saving throws against spells and other magical effects.
This way of defining special traits with an ad hoc rule particular to that creature allows the designers and the DM significant flexibility when creating creatures. It also means that you won't find a general explanation of how magic resistance works outside of a creature's entry, because it's individual to each creature.
Although magic resistance works identically for most (all?) other creatures in the MM that have it, defining it per-creature in 5e means you can't rest comfortable knowing how a monster you're facing works: DMs have the power/authority to alter creatures' special traits, and new creatures (either new-to-your-PC, or in new books) can be made that have a "Magic Resistance" with a different definition.
A good example of this way of using special traits to define traits unique to one creature is the Rakshasa's magic immunity:
Limited Magic Immunity. The rakshasa is immune to spells of 6th level or lower unless it wishes to be affected. It has advantage on saving throws against all other spells and magical effects.
It specifies which spells it's immune to and when. You can probably see how the flexibility inherent in this design approach could be used to give a different creature a trait with the exact same "Limited Magic Immunity" name, but with a different description that makes it immune to only Illusion spells, or spells of 3rd level or lower, or spells cast by a cleric, etcetera.
Additionally, see how this gives a magic resistance effect identical to the Death Knight's Magic Resistance, but without the Rakshasa having a Magic Resistance trait! It's just part of the ad hoc definition of its Limited Magic Immunity trait.
No, a Warlock with the Pact of the Chain feature does not receive Magic Resistance if they choose a Quasit, Imp, or Pseudodragon. In the first place, a variant rule is only in play if the DM chooses. However, even if the DM decides that pseudodragon familiars (for example) are a thing, it still doesn't benefit the Warlock.
Some pseudodragons are willing to serve spellcasters as a familiar. Such pseudodragons have the following trait.
So "some pseudodragons" will be familiars who share Magic Resistance with their masters. The Warlock, however, gets their familiar from the find familiar spell, which says:
You gain the service of a familiar, a spirit that takes an animal form that you choose: [...]
The Pact of the Chain says that:
When you cast the spell, you can choose one of the normal forms for your familiar or one of the following special forms: imp, pseudodragon, quasit, or sprite.
In other words, the Warlock's familiar isn't any sort of pseudodragon, let alone one with the variant. It's a spirit that takes the form of a pseudodragon (and can take a different form anytime the Warlock chooses). The spell does say that:
[...] the familiar gains the statistics of the chosen form [...]
However, the statistics of a creature are defined on pages 6-11 of the Monster Manual. The short definition is that, if you look at a creature's entry, it's everything in the yellow box. This doesn't include variant traits like the Familiar trait - those are in green boxes off to the side.
All of that aside, a generous DM could, of course, allow a Pact of the Chain familiar to provide its master with Magic Resistance. It must be pointed out, though, that this is making the Pact of the Chain vastly more powerful than it normally is, so DMs thinking about it should consider carefully.
An even more generous DM could allow any player to gain the service of a creature with the familiar variant. This, too, should be considered carefully - it's effectively giving the player a more powerful version of the Pact of the Chain feature for free.
Best Answer
It's really good, but not necessarily game-breakingly so.
It's rare for a player character to get advantage to something in all circumstances like this, but "spells and other magical effects" is only a small subset of dangers a player character tends to face. It also only applies to those effects which would normally grant a saving throw.
The Sage Advice Compendium, in answering the question "Is the breath weapon of a dragon magical?", suggests that the game's designers intended "magical" to be more limited in scope than it may seem. For example, a red dragon's fire breath is explicitly not a magical effect. "Magical" is defined as any one of the following:
The majority of monster effects, even those which would clearly be considered supernatural, do not fall under this definition of "magical". The Sage Advice Compendium document is considered to present official game rulings, though a broader understanding of "magical" as it relates to monster attacks would make this ability more powerful.
Magic Resistance would protect you from the aboleth's Enslave ability, a planetar's Innate Spellcasting, a basilisk's Petrifying Gaze, a beholder's Eye Rays, a death knight's Hellfire Orb, a dryad's Fey Charm, an empyrean's Bolt (defined as a ranged spell attack), or any spell cast by an NPC spellcaster which allows a save.
However, it would not protect you against a red dragon's breath weapon, an ankheg's acid spray, a banshee's wail, a behir's Lightning Breath, a demilich's Howl or Life Drain, a balor's Fire Aura or Death Throes, a fire elemental's flammable nature, a ghost's Possession, a storm giant's Lightning Strike, a night hag's Nightmare Haunting (it's magical, but allows no save), a hell hound's Fire Breath, or any spell which does not allow a save.
It would also not protect you against melee, ranged, or natural weapon attacks, which are often the primary source of danger. Most monsters don't rely on magical attacks for their primary effects (for example, the hell hound's fire breath is only a recharge 5-6 power), and many monsters have no magical attacks at all.
In summary, magic resistance is really good when it applies, but in the hands of a player character it typically applies less frequently than you'd think.