Zapdos is an incredibly good Pokémon, but its main counters, Golem and Rhydon, are on a vast majority of RBY teams, thus limiting its effectiveness. It does a paltry 10% to these two with Drill Peck, and it cannot hit them with any of its Electric moves. The best way to use Zapdos (along with almost any other Electric) is to hide it for as long as you can and wait for Golem/Rhydon to be removed; then, and only then, should you show your Zapdos. As a late-game sweeper, Zapdos is nearly unrivaled because of its amazing mixed sweeping capabilities.
This set takes full advantage of Zapdos's superb stats. It has an amazing Special backed up by a very good Attack, and combined with its decent Speed, it can and will cause some havoc. The problem lies in its inability to hurt Rock / Ground types such as Golem and Rhydon, which happen to be on nearly every RBY team. Remove these threats and Zapdos can wreck opposing teams.
Thunderbolt coupled with Drill Peck will cause some very good damage to anything not named Golem or Rhydon (or any other Rock / Ground). Thunder Wave can slow down any incoming Pokémon so they can be easy pickings for Zapdos later on in the match. The fourth move is a wild card and completely up to the user. All of these moves have their advantages but not one outshines the rest.
Agility causes Zapdos to become faster than everything else and ignores the Speed drop that comes with paralysis (Zapdos can still become fully paralyzed, however). Rest removes any status and helps to keep Zapdos alive, but can be tough to pull off, and Zapdos would be better off attacking. Thunder can OHKO Starmie and Slowbro without a critical hit, and will 2HKO Tauros, but only has a 70% chance of hitting. Mimic is used to steal any incoming Golem or Rhydon's Earthquakes, thus letting Zapdos damage these two. Unfortunately, Mimic has some inherent flaws that limit its effectiveness. It only has a 25% chance of copying Earthquake and even then it can only do about 30 to 37%, effectively a 3HKO, while Golem and Rhydon will 2HKO with Rock Slide.
Hyper Beam is a good option for late-game sweeping, as it does more damage than Drill Peck to most Pokémon, but be sure it will finish off the opposing Pokémon as Zapdos does not like sitting around on the recharge turn. Toxic can be used to hit Golem and Rhydon on the switch and making them take damage every time they do switch in. Reflect and Light Screen can be used to tank, but are not recommended on the same set together. Coupled with Rest, Zapdos can stay alive for quite a long time.
Checks and Counters
Golem and Rhydon are its best counters; Zapdos cannot hurt either of them and they can do whatever they want as Zapdos will be forced to switch or be 2HKOed by Rock Slide. Chansey can come in on Thunderbolt and take very little damage, and is only 3HKOed by Drill Peck. This lets Chansey use Thunder Wave or Ice Beam and then Softboiled to regain its HP. Be wary, though, as one untimely critical hit can ruin Chansey. Other Electric-types, such as Jolteon and Raichu, will take very little damage from either of Zapdos's attacks and will hit back with a neutral Thunderbolt for a decent amount of damage.