Specialized in Air-To-Ground combat, Bombers are capable of devastating most any ground or sea unit in one attack.
Bombers are the counterpart to Fighters; Bombers deal very high damage to Ground units, but are unable to attack Air units. Full-HP Bombers will outright destroy most Ground units in one attack, or leave them with minimal HP. Even Mega Tanks, which are normally very resilient units, take moderate damage from Bombers. As such, Bombers are great units for breaking defensive lines and exposing an opponent's more valuable units to follow up attacks.
As stated above, Bombers are completely helpless against Air units. B-Copters cannot attack Bombers, but they can block a Bomber from hitting more valuable units. Fighters and Stealths will also make quick work of a Bomber, so care must be taken when those units are on the field. Just as well, Anti-Airs will also do moderately high damage to a Bomber, but only if they can get the first strike; Bombers are just as deadly to Anti-Airs as Anti-Airs are to Bombers. With their higher movement and lack of terrain movement penalties, Bombers can easily outmaneuver Anti-Airs when used correctly.
Bombers excel when near full HP, as Bombers below 9 HP will lose most of their One-Shotting potential; severely hindering their ability to open up the path of attack for other units. It may be advisable to invest in repairing damaged Bombers where possible.
Anti-Airs are typically the most readily available unit around to combat enemy Bombers, though since they are slower units, are more used as a deterrent to Bombers as opposed to being an aggressor. Position Anti-Airs near the rest of your force so that any potential Bomber attacks would be met with a retaliation attack from an Anti-Air. Be sure to keep your Anti-Air itself protected though; if exposed, the Bomber can just attack the Anti-Air, neutralizing it in the process. By the time Bombers are being built, Stealths and Fighters will also likely be readily available to build, both of which can give Bombers a lot of trouble and restrict safe airspace.
When an enemy is fielding Bombers, avoid placing units on Roads and Plains where possible. Units on those terrain features will likely be destroyed in one Bomber attack, leaving more valuable units vulnerable. Units on Woods, Properties, and Mountains will still take very high damage from a Bomber, but will likely survive the attack with minimal HP, keeping your defensive line intact for a turn.