This Arcane Tradition is dedicated to learning and understanding magic and it's very nature

Analyze Spells

As a reaction, you can observe any spell being cast and learn it. Roll an Intelligence check adding your Wizard level, vs a DC of the observed Spells Level +15.. With success you were able to analyze the spell, components, movement, etc, and break it down. You can copy it to your spellbook on your next long rest at the usual cost to copy spells to your spellbook. This can be done once per long rest. The spell needs to be an arcane spell, but does not need to be on the Wizards spell list, though it becomes a Wizard spell for the wizard. 

Arcane Skill

You have advantage on Arcana checks and History checks related to magic. 

If your campaign requires any sort of check to learn a new spell, then you have advantage on those checks. 

Arcane Resistance

This gives you a +1 bonus every 3 levels against spells cast against you. This could be a bonus to the saving throw vs the effect, or some other bonus depending on the type of spell. Note that this is only a defense against the magic itself. Not against things caused by magic. It will not for example lower damage from fire, since fire is not magic, even if it is created by magical means. This only works against spells that allow some sort of defense, such as a saving throw to resist it. If the spell does not allow a save or other chance to resist it, this feature has no affect against it. 

Arcane Sense

The character rolls a perception check to sense the presence of arcane energies. This can be represented by seeing an aura, touching an item and feeling the energy, just a feeling inside, or even smell or taste. Stronger or higher levels of magic will be easier to sense. The DM may provide more information, like the type of energy or other information, depending on the level of success. 

Familiarized Spell

One spell of the players choice that the character knows already is always known and doesn't need to be prepared. They can cast it at any time by expending a spell slot of 1 level lower than the actual level of the spell or higher. If the spell has a cost listed component that will need to be available and spent. Otherwise, it will not require a material component or focus.