Starting at second level, wizards can choose a school of magic to go into. Formally, there are eight schools, but it can be argued that there are 9: abjuration (protective), conjuration (transportation), divination (revealing information), enchantment (mind spells), evocation (magical energy), illusion (decieving minds), necromancy (life/death energy), transmutation (changing properties of matter) and the unofficial school of invention.
Wizards can get certain benefits depending on the class they choose to go into. For example, if a wizard chooses the school of evocation, they get the benefit of having to spend less time and gold on spells. You also gain the ability to save certain creatures from being hit by spells, make creatures take damage from a spell even if they succeed on a saving throw, add your intelligence modifier to the damage of any spell you cast, and force certain spells to do maximum damage every time they are cast. Each of these bonuses are added at a different level.
In D&D beyond, there is also one other school that wizards can choose: the school of invention. This school boasts that it invented all other schools of magic, a claim which most do not believe. In this school, at 2nd level alone you gain proficiency with 2 tools of your choice, a free suit of armor, and Reckless Spellcasting, which allows you to cast spells chosen from a list of spells without any preparation. At 6th level, you gain the ability to channel your magic through your free suit of armor, allowing you to manipulate the damage type and damage amount. At 10th level, you can replace a spell that you have prepared with another spell from your spellbook, and at level 14, you can improve your Reckless Spellcasting.