Budgies are flock animals. They naturally live in huge flocks in the outback of Australia. Many people are against keeping a single pet bird, others justify the fact that one budgie can be equally happy.
If you decide to keep one budgie, it naturally has it's advantages. You don't have to buy as much food and you can usually purchase a smaller cage than what you would have to buy for heaps of parakeets. The noise level will be considerably lower, as two or more budgies stimulate each other to make a bit of a racket. Obviously, you will not have to worry about baby budgies, but females will still lay unfertilized eggs.
If you buy one budgie, you are much more likely to succeed in teaching it to talk (for information on this, there are some helpful threads on this site)
One budgie will also become more tame as a general rule, because their flock is humans and humans are the flock. A single budgie will often learn tricks easier, and constantly amuse you with its attempts to impress you by swinging on bells and climbing up ropes...
If you are out at work or do not have the time to play with your budgie often, it will grow lonely and sad without company, and will usually not talk or become tame. In this case you are better off getting it a companion.
Two budgies will generally be nosier, and are usually less reliant on human contact. They will thrive on each others company, and will take longer to tame, as they will not feel the need to fit in with humans. Budgies find it easier to form one strong bond (with their parakeet companion). Many people have two budgies that love attention from humans, but still prefer each others company.
This is generally speaking, and each budgie (or pair of budgies) varies.
If you get a male and female, they will usually mate and lay eggs. If you do not provide a nest box, the eggs will not hatch.
Two females together can be a disaster, although sometimes females may get along, it is advised you get a pair or two males.
As for talking, two budgies will prefer to use budgie talk, and do not really feel the need to communicate with humans. But if you put the work in, you may get two budgies to talk, although I would not guarantee success.
Two budgies are a good choice if you are at work or don't have time to spend with them.
More than two budgies:
If you want to have lots of budgies as pets, they can be very noisy in comparison to one or two budgies.
You will be much less likely to develop a bond with them, because in their heads they have their flock and don't need some giant human trying to join it. However, as with everything related to budgies, patience will pay off, and you might just get them tame and friendly.
Lots of budgies will rarely or never learn to talk, unless you are prepared to put in the effort to separate them for every single talking session. It would be very hard, but your birds may surprise you one day!
More than two budgies will be fun to watch as they squabble over perches, and as they kiss and preen each other.
Lots of budgies can also be very, very, messy!
More than two budgies is a good choice if you cant spend much time with them.
If you want to have aviary birds, they will usually not become tame. If you are looking for a pet, aviary parakeets will be lovely to watch but it is unlikely you will bond with them. I have never have aviary budgies so I am not exactly and expert. Maybe someone else could help you a little more with this one.
I cannot say how each budgie is suited to each different way of life. Some budgies thrive on their own, while others long to be with others of their kind.
Remember this: patience will pay off. You will only get good results from you budgie/s if you have the patience to train them.
Good luck. Whatever amount of parakeets you choose, you are guaranteed to get many years of joy from these little creatures with such a lot of character!