Nidoking is better suited for most competitive teams because of his higher Speed and offenses, in addition to Megahorn, all of which let him better pulverize opposing teams. However, Nidoqueen is much better at using Counter due to her greater bulk. Nidoqueen also more effectively takes advantage of her resistances, but you should still be wary of these nasty weaknesses to Earthquake, Psychic, Ice Beam, and Surf.
This set is usable in standard play. Superpower OHKOes Tyranitar and does massive damage to Snorlax, Blissey, and Regice.
Thunderbolt, Thunder, and Fire Blast are all options in the last slot to take out Skarmory; shift 56 EVs from HP and Speed into Special Attack and use a Hasty or Naive nature in order to nab the OHKO on Skarmory.
Counter is an awesome move on Nidoqueen as she can be EVed to survive even STAB Earthquakes. Charm can annoy physically based switch-ins, while Roar counters Baton Pass teams, as well as Ninjask and Scizor. Nidoqueen is defensive enough to pull off a RestTalk set, but it's better to capitalize on her outstanding versatility. Substitute helps against Weezing if it's not running Flamethrower or Fire Blast.
Aerial Ace or Hidden Power Flying are options to destroy Heracross; in the same vein, Hidden Power Grass 2HKOes Swampert. However, Nidoqueen must hit it on the switch or be KOed by one of Swampert's STAB moves. Crunch is an option for a specially based Nidoqueen to damage Gengar, but note that such a set is not recommended as it lacks STAB. Double Kick makes Nidoqueen a good Tyraniboah counter as it breaks Tyranitar's Substitute and damages it in the same turn; furthermore, Nidoqueen resists Focus Punch and is immune to Thunderbolt.
Checks and Counters
Bulky Water-types, such as Suicune, Milotic, Slowbro, and Swampert, are failsafe switch-ins. Their high HP and Defense stats enable them to easily switch in on even Thunderbolt and Choice Band Earthquake. Swampert has more trouble with Hidden Power Grass variants, however. Nonetheless, all of the aforementioned Pokemon pack super effective STAB moves to take down Nidoqueen. Weezing is a good counter to Nidoqueen: in addition to possessing a high Defense stat, it is immune to Earthquake, resists Focus Punch, Superpower, and Sludge Bomb, and packs Will-O-Wisp to further lower Nidoqueen's attacking power. Substitute variants can be a hindrance, however, as Substitute protects Nidoqueen from Will-O-Wisp, and Nidoqueen's Substitutes survive one Sludge Bomb. While Focus Punch will put a dent in it, Skarmory takes most of Nidoqueen's physical attacks easily, thanks especially to its immunities to Sludge Bomb and Earthquake. As long as Nidoqueen doesn't pack a Fire or an Electric move, Skarmory can easily lay Spikes, phaze Nidoqueen away for some token damage, or wear it down with Drill Peck. Forretress isn't as easily killed by Thunderbolt or Thunder, but is OHKOed by Fire Blast. However, it can damage Nidoqueen more easily with super effective Earthquake, so it's something to watch out for.
Flygon can come in on Earthquake, Rock Slide, Sludge Bomb, as well as Fire and Electric moves; in fact, it only really has to watch out for Ice Beam, which might not always OHKO. In return, it can badly damage Nidoqueen with Earthquake, which OHKOes if Flygon is equipped with a Choice Band. However, a fully defensive Nidoqueen can survive non-Choice Band Flygon's Earthquake and KO it with Counter. Specially defensive Claydol barely takes 30% from Ice Beam, and can eliminate Nidoqueen with an Ice Beam of its own. It also packs STAB Earthquake, but must watch out for Counter variants. Donphan is 2HKOed by Ice Beam and weakened variants should therefore not switch into Nidoqueen, but aside from that, it makes a nice counter due to its high Defense.
Celebi resists Earthquake, but is weak to Shadow Ball and Sludge Bomb. Nonetheless, it can shrug off most non-Choice Band-boosted attacks, especially with Recover, and hit Nidoqueen hard with STAB Psychic. Finally, Dugtrio can come in on Thunderbolt, Fire Blast, or Sludge Bomb, on a double switch, or after a kill, and proceed to OHKO Nidoqueen with STAB Choice Band Earthquake.