Of course this doesn't apply to right now since no one is allowed to travel from country to country but for ages you were and I think one day again soon, possibly after some elections are over, things will start to open back up again.
When I first traveled to SEA (South East Asia) there weren't any budget airlines but then the next several times I returned there were loads of them and I will detail a few of the better ones here.
Nok Airlines
Nok just means "bird" in Thai and it is as appropriate of a name for an airline as any other I suppose. Nok airlines impresses me because even though it is no frills there seems to be a company culture of friendliness that doesn't really exist in other budget airlines or even far more expensive full-service airlines.
They are always near the cheapest (or THE cheapest) option when choosing flights around Southern Asia and even if I have to pay a little bit more, I will choose them if they are an option.
They almost have a joking manner to the way that they operate and since air travel can be quite stressful anyway, I tend to appreciate that sort of humor even if some people don't think it belongs there.
The only downside of Nok is the fact that they are very small and don't operate to very many destinations. They primarily operate out of Thailand and a few other international locations.
Bangkok Airways
Bangkok Airways will always cost around 10 - 20% more than Nok but they aren't technically a budget airline - they are simply a full-service airline that is much smaller than the other full-service airlines but the only way they can compete is by having lower prices.
What makes Bangkok Airways so special is that at larger airports they have their own "luxury" lounge that is only available to Bkk Airways ticket holders. You do not need to be in business class either... it is available to all the poor people back in steerage as well. For anyone that has ever had to wait for a few hours for their flight to depart this is absolutely brilliant because they have lots of comfortable chairs, their own bathrooms, free food and drinks, as well as a reminder board that details ONLY BKK Airways departure times.
In my mind, it is totally worth the little bit of extra money.
Air Asia
The grand-daddy of them all, Air Asia got the ball rolling in SEA many years ago and they are by far the largest and have many times the amount of destinations as the other two as well as others that I did not mention. They tend to have the lowest prices of anyone for just about any destination. It is very rare that any other company can compete with them on price and this is their entire business model.
The only downsides of Air Asia is that somehow they ended up being the least friendly of all the budget airlines and they even nitpick you on various charges such as baggage allowance and wont let you have so much as a bottle of water on the plane without paying for it.
They define "no frills" because honestly, you are going to get nothing other than the flight when you book with them. If you are carrying more than 7kg of luggage you are much better off flying with one of the other airlines because after the luggage fee is factored in (and it is extortionate) it is no longer the cheapest of the bunch. However, there are many instances where they are going to be the only choice as some flight routes are only served directly by them. I have no idea how this works but it does seem to be the case.
Of course if you are backpacking I highly suggest that you investigate traveling by train or even by bus provided it is significantly cheaper. In the case of 2 of the above 3 it has been my experience that the airlines in many cases are actually cheaper than ground transport and I can't imagine how that is possible, but it does seem to be the case.