With so many travel tools available to us now, the best way to get the best airfare to suit your needs is to do your research.
Research various airlines
Personally, I like to use websites that show all of the different airlines' fares on the one page but seldom actually use them for my booking. The reason for that is that they will all add their own booking fees so that airfare that you thought was really cheap will end up more expensive.
So, tip number one would be to use comparison websites to do your research but don't book with them.
Be flexible with your dates
Also, if you are able to do so, be flexible with your dates - try playing around with mid-week dates instead of weekends. This sounds silly but fares will differ dramatically if you are able to leave and return on a Tuesday or a Wednesday as opposed to leaving on a Friday night and coming back on a Sunday night (or even in some cases Monday morning).
Sign-up to airlines' mailing lists
If you are planning a trip, I can't stress highly enough how important it is to be on different airlines' mailing lists. This way, you will be the first to know about any promotions that they may have. These promotions sometimes only last a few days so it's best to know about it as it launches so you can take advantage.
Virgin Australia, when it was known as Virgin Blue, was the first airline in the Australian market to introduce the happy hour special concept. Basically, between certain hours each day, certain routes with a specified travel period would be reduced. In the case of Virgin Blue, you had to book within that one or two hour window, otherwise the fares reverted back to the original pre-happy hour fare.
The Virgin Australia Happy Hour is on from 4pm - 6pm Australian Eastern Standard Time (AEST) on weekdays.
Jetstar have a similar promotion, known as Friday Frenzy which is every Friday afternoon (as the name would suggest). If you sign up to its mailing list, Jetstar will notify you of the Friday Frenzy fares (wow, there are a lot of Fs there) by email each Friday.
Be flexible with your cities
This obviously applies for international rather than domestic travel. In the past, I have been able to take advantage of ridiculously cheap airfares to Europe by being flexible with my cities. When Qatar Airways launched its Brussels, Belgium route, I was able to fly from Melbourne to Brussels return for A$793. Even though I was living in Sydney and not in Melbourne, I still managed to fly to Europe return for under A$1,000 on a premium airline (Qatar Airways won the award as Skytrax World's Best Airline for 2011 (the year I flew) and 2012).
Similarly, even though I was actually overjoyed to be able to fly straight into Brussels as it meant I could go down memory lane (I lived there for a year in 2002), many who were on the flight with me told me that they were using it as a launching pad into all of the other cities on the doorstep. From there, some were able to take the Eurostar directly into the heart of London.
Become the rope in a tug of war for your business
About 10 years ago when I was looking for fares to go to Germany with about one month's notice (and back when it wasn't quite as easy to look at all the different airlines' fares as it is now), I asked two different travel agencies for quotes. Travel Agency A ("A") was an online only travel agency and Travel Agency B ("B") was an agency from a chain with many stores. I didn't act on either of the quotes straight away and within a few days I had A emailing me asking if I was going to take the fare. I replied and told them that B had quoted a better fare. They then asked to see the quote and proceeded to offer me a better price. Then B asked me if I was going to take the fare they had quoted me and I told them that A had bettered their fare. B then said that they would better A's fare. Inadvertently, I had become the customer that both A and B wanted to win. I ended up going with B in the end and they told me that they were actually only making $1 on the fare by the time it had been bettered by A then B then A again and finally B.
Flight Centre is an example of an airline that has a best price guarantee. It will better the fares by $1 but unfortunately no longer gives you a $20 voucher for use on domestic bookings. This might not sound like much but if you end up in a bidding war like I did, it may work to your advantage.