In the wild, a budgie's aggression serves important purposes, like securing territory from rivals or ensuring a more secure spot in a group hierarchy. In captivity, that same aggression can become a problem. However, if you understand what's triggering aggression in your bird, you can often diffuse it.
Jealousy
Budgies are very social animals. When a new bird is introduced, they often feel threatened. If your budgie is most aggressive toward a new bird and is more vocal in your presence after adding a new budgie, there's a good chance the aggression is motivated by jealousy. You can defuse this situation by making a point of giving your entire flock lots and lots of attention.
Basic Incompatibility
This form of aggression is a bit more passive than others. Some birds just don't like each other. In this case, you would see your birds consistently stay on the opposite sides of their cage and refuse to engage in grooming behaviors with one another. In this case, you should separate the birds and provide them each with mirrors to keep them entertained.
Breeding Behaviors
Breeding can trigger all sorts of aggressive behaviors in budgies. If you have two birds of about the same size who do everything together (vocalize, groom), they're probably getting ready to breed. Birds in this state will start to nest and aggressively defend their nesting site from other birds. You can short-circuit this process (and the aggression) by only allowing your budgies 10 to 12 hours of daylight and not having any dark, recessed areas where they can start to nest.
Territoriality
Sometimes birds just don't feel like they have enough space or resources. This type of aggression will look like birds "defending" feeding areas like bowls or a favorite perch. These behaviors will be most likely directed at new birds. If you see this sort of behavior, consider a larger cage, multiple sets of food/water dishes or even splitting the birds into separate cages.