While Articuno has a lot of counters, it is truly underrated. It can Roar away the feared Calm Mind Celebi, laughs at most Swampert, and can easily stall out Blissey and all sorts of Water-types. As Grass-types don't appreciate the threat of a STAB Ice Beam, Articuno does a great job at absorbing Sleep Powder and Spore, and even Milotic's Hypnosis. Additionally, Articuno nullifies Snorlax with Reflect and either Haze or Roar. It also scares away any Salamence without Rock Slide, and keeps it from coming in regardless of the set. Finally, Articuno is not 2HKOed by most non-STAB Thunderbolts. It takes some prediction to do good damage with Articuno, though.
Articuno is a stall machine; with good HP and Defense, and marvelous Special Defense, it makes a premier Sleep Talk user. As the real strategy of this set is to rack up as much Spikes damage as possible, make sure you have that on your team when using Articuno.
Articuno should mainly look to come in on Celebi and Swampert. If the latter lacks Rock Slide, neither can hurt Articuno much, and they will simply waste their PP to Pressure. Your opponent will probably realize this and switch out; this is where you outpredict and use Roar. Spikes will deal damage to both your opponent's counter of choice and the random Pokemon dragged in by Roar, provided they aren't Levitate users or Flying-types—most of whom are hit super effectively by Ice Beam. Additionally, because they would likely have switched to their counter, their active Pokemon will often not be their best answer to Articuno. The real draw to this set, though, is when Articuno is sleeping. If Sleep Talk selects Roar, and Articuno happens to be faster than its opponent, Articuno will Roar the foe away before it can do anything for free Spikes damage.
In a nutshell, this set is pretty simple. Stall to eternity and back, come in on Water-types, Blissey, or the aforementioned Swampert and Celebi to safely Rest up, and make sure not to run out of Ice Beam PP (Sleep Talk helps with this).
Articuno gets Heal Bell via Pokemon XD, and can be a decent cleric if it needs to. It's not as good in this role as Blissey, Celebi, or Miltank is, because it lacks a 50% recovery move, but it lasts long nonetheless. Again, one can choose a RestTalk strategy, which also helps conserve Heal Bell PP. Reflect supports the team and guards Articuno against Metagross and users of Rock Slide. Toxic cripples Tyranitar, Snorlax, and Suicune, but is useless against Steels and Blissey. Articuno can make use of Roar, of course, though not as effectively as it can in the first set.
Always maximize Articuno's HP for optimum Leftovers recovery and maximum durability. Use a Calm nature as Special Defense is Articuno's greatest stat and should be boosted. As Articuno has the same Speed stat as Suicune, it should run a small amount of Speed EVs to Roar Suicune out first; this also guarantees Articuno outruns Jolly Marowak. If you're not using Roar, this obviously becomes less important, but beating Marowak is still useful, and you might want to get in a Haze before being hit by Surf. The spare EVs can go into either Special Attack or Special Defense.
This set is designed to sap as much PP as possible from the opponent, but a smart opponent will just switch his way out of it. Articuno's Substitutes can survive STAB Surfs, which is very nice. The fact that Meteor Mash and Rock Slide have a maximum of 16 PP helps Articuno a lot.
The only reason to use Haze over Roar is because the former doesn't have negative priority, but Roar's ability to rack up Spikes damage is generally preferable. Hidden Power Fire lets Articuno take down Metagross, and Hidden Power Flying does a number on Heracross.
Checks and Counters
Rock Blast Octillery makes the best Articuno counter because it's immune to Roar and resists Ice Beam. On a more serious note, Metagross resists Ice Beam and can crush Articuno with STAB Meteor Mash. Tyranitar takes neutral damage from Ice Beam, but knocks out Articuno with Rock Slide. Raikou and Jolteon take light damage from Ice Beam and 2HKO with Thunderbolt. Rock Slide Swampert, though uncommon, is easy to bring in and will usually 2HKO Articuno. Any Fire-type with decent Special Attack puts the hurt on Articuno as well.
Registeel does not threaten to OHKO, but can wall Articuno all day, and can gradually wear it down with Rock Slide or Hidden Power Steel. Regice might not be able to KO Articuno quickly with Thunderbolt either, but doesn't take much damage from Ice Beam; the same applies to Lanturn, Lapras, and Starmie.
Snorlax takes next to nothing from Ice Beam, can boost up with Curse, and hit Articuno with STAB Normal moves before it can Roar Snorlax away. Jirachi resists Ice Beam, can set up Calm Mind boosts, and hit Articuno with a super effective Fire Punch, Thunder, or Thunderbolt. Blissey can wall Articuno, but will not win a one-on-one stall war thanks to Pressure.
In general, anything that can take an Ice Beam and use a Rock move is a great counter. However, it is worth noting that all of Articuno's counters are susceptible to Spikes.