Burna Boyz are Orks who cannot stop setting things on fire. These lunatics will set light to anything and anyone for the simple joy of watching them "do da burny dance." They can prove utterly lethal to tightly packed or lightly armoured infantry, the roiling blasts of their Burnas flushing their victims out of cover amid the sizzle.
Often they join forces with Mekboyz, whom they depend on to supply them with Promethium fuel and the Burna weapons from which they get their name. These dedicated arsonists take great delight in setting fire to the enemy (and anybody and anything else they can lay their hands on if no enemy is available), and it takes a strong Warboss or the blackmailing powers of the Meks to keep them in line. However, the benefits of these lunatics in any Ork warband just about outweigh the risks of getting anywhere near them. The Burna Boyz can be a powerful asset on the battlefield, whether in a small team or large mobs; their mass incendiary firepower can prove devastating, particularly against enemy infantry, flushing out foes from cover and bringing down bestial creatures. Also, with their weapons turned to the searing hot point of a blowtorch flame, they can wield them to hack and slice up enemy armour and vehicles. They serve a vital secondary role in this regard, cutting up battlefield salvage for later use.
Burna Boyz are an elites choice, meaning that they compete for a slot with with Tankbustas, Nobz and Kommandos. As a result most Ork players don't bother looking to put them in their army lists half the time -- the Ork codex is and always has been crawling with anti-infantry damage, Burnas feel redundant when their slot could be better spent on a different, less overflowing niche. This is not to say they're bad, they're actually a very good anti-infantry unit, especially in armies revolving around the use of many vehicles. As you can imagine every Burnaboy is armed with a weapon that counts as a Flamer in the shooting phase, and a melee weapon with the relevant edition's power weapon rules in the Assault phase (for example, AP3 in 6th edition).
Versatile Burna - Burnas can be used as either a flamer or Power Weapon, which means they can attempt to cut down MEQ units or Monstrous Creatures (although they still suffer from a crippling lack of strength for this task). However, the choice must be made whether to use them as such during overwatch, if attacked first. If you use the d3 auto-hits that means you cannot use them as power weapons. Considering the average model is MEQ, overwatching on d3 is better as the chance of an unsaved wound is the same chance as hitting and (in 8th edition) wounding at AP -2 in CC (plus 2A extra per boy in CC if overwatching). Per model basis now means that you can elect to fire with half of the squad while the other half preps power weapons.
Dedicated Transport - Burnas can now take trukks as a dedicated transport. Remember that when you fire from a Trukk, you can simply find one good placement for the flamer and have every single Boy shoot with exactly the same angle resulting in dozens of hits on a same unit.
Through judicious use of mass tank-shocking (pre-8th edition) or Advancing (8th edition), enemy infantry units can find themselves surrounded quickly, making them prime targets for being purged with fire. The ability to unleash 30, 40, even 60 strength 4 attacks which automatically hit is frightening: don't be surprised if your opponent just removes the squad and skips saves.
But then we get to their drawbacks. In addition to the aforementioned Elite slot overcrowding, they are a slow unit heavily reliant on their transports to get anywhere... except they cannot fire from transports if you move the vehicle any great distance. As soon as their transport is destroyed, the enemy can simply retreat slightly and continue firing at the expensive Burna Boy unit that only has Toughness 4 and a 6+ armour save to protect them. Yeah, Burna Boyz are one of those "expensive gun, fragile soldier" units like Melta Space Marines or Guardsman Veterans, meaning they are entirely reliant on transportation and thus mobility to ensure they come anywhere close to proving their worth in an Ork force. This was not helped by the fact that they were themselves very weak to Flamers in certain editions (namely 7th) if they rode in an open-topped Trukk, or their base Initiative 2 if they tried to actually use those fancy melee rules.
8th edition dropped their points, for a very good reason: Burnas were nerfed from being exactly as strong as an Imperial Flamer, to half as powerful. A Flamer deals D6 hits, a Burna only D3, a change that was made rather late in 8th edition's development. Based on Mathhammer it's estimated that a Burna costs 4 points to a Flamer's 7. The upside is that a Burna Boy functionally acts as if it had a Power Stabba (S:user, AP:-2) for free, even if they only have 2 attacks base. A lot of players are salty about this nerf, and overall it has made taking them even harder to justify even though they're relatively cost-effective as long as they actually stay alive (see above).
Regardless of edition, Burnas are generally a tricky unit to use effectively, although cheap 5-man squads with a Trukk in MSU lists can be very effective infantry hunters, backfield objective clearers and even distractions through their insane damage output against other Xenos infantry.