The way budgies warn each other off is usually through body language and threatening behavior. The budgie will try to make itself appear big by standing up straight and extending all its joints. Then the budgie will sound a warning call and maybe hack towards the offending budgie.
Now, as far as I know, there hasn't been a single recorded case of budgies fighting in the wild. And I can't blame them. They have to work hard enough to survive in the desert like conditions without fighting with each other.
There are a fair number of recorded cases of budgies fighting each other in captivity though. It's probably a result of having to live so close together all the time. If two budgies see each other as threatening their survival for whatever reason, and they're stuck in a cage or aviary together, then fighting is probably inevitable.
Two budgies that have bonded and accepted each other won't fight. They'll be happy living together and respect the other's need for space. And male budgies will almost never fight females.
Female budgies are usually more aggressive than males. Conflicts will often happen when there are more female than male budgies in a group. The females tend to see each other as rivals, even if the male budgie is already bonded with one of them. So, make sure that you either have equal numbers of male and female budgies or more males.
The other time that fights are more likely to happen between pet budgies is when they want to mate. You can avoid aggressive competition over nesting sites by making sure that there is plenty of choice of nesting boxes. And by making sure that the nesting boxes are spaced out at least three feet apart. If the nesting boxes are too close together both pairs of budgies will feel threatened by the other pair.
Budgies aren't really equipped that well for fighting. They don't fight their predators in the wild, they just run away. So, when budgies do decide to fight all they've got as a weapon is their beak. And it's not a particularly sharp beak compared to a lot of other birds.
When they decide to fight, one budgie will approach the other and try to bite at its feet and pull its feathers. The other budgie will try to use its beak to block the attacker. The fight usually lasts until one of the budgies decides to run away. It's very rare indeed that one budgie will continue attacking after its opponent has decided to flee.
If that happens all you can really do is try to separate them before there is any serious injury.