A budget is how much investments (ie upkeep) you spend on a building.
In general, the higher the budget the higher the Effectiveness of the buildings. In other words, production buildings and factories will produce more for the same amount of time while welfare and tourism buildings gives higher service quality and in some buildings will also allow them to provide their services to more people.
In addition of increasing the building effectiveness, increasing a building's budget will also increase the workers' wealth, which gives them access to more expensive housing and entertainment.
Keep in mind that increasing the budget of housing buildings will also increase their rent costs, which means it will increase the wealth level required for their access, or in other words doing so will block less wealthy citizens from using them.