I'll be editing this as I think of more ideas. Feel free to provide input.

Getting just one hero to counter one other hero will not win a game, but picking a hero to fight against an opposing heroes strengths can be beneficial to a team. Team coordination is more important than counter-picking, but here are some ideas on who to pick against who. Remember that just because you picked a counter doesn't automatically mean that you win.

Death Prophet < Lifestealer
Death Prophet is dependent on her ultimate and her tank-ability in most cases. She will pop off her ultimate, which deals a ton of damage as well as heals her. Lifestealer deals damage based on a heroes max health, and has lifesteal, so he will be able to walk through her spirits and regain life upon hitting her.

Huskar, Pudge, Dazzle < Axe
Axe can take a lot of damage and his ultimate will allow you to deal an instant killing blow to anyone with hitpoints lower than 300/450/625. His ultimate also goes through shallow grave. Huskar and Pudge are heroes that typically run with low health. Huskar to aid his damage output and Pudge because rot damages himself.

Anti-Mage < Faceless Void or high HP and armor heroes
Faceless can, one on one at the same level, outlast AM head to head. He has both a bash ability, one that allows him to evade attacks, and an ultimate that allows him to lock targets down and deal damage. AM only has a magical resistance buff, which is useless against Faceless, and his ultimate is dependent on how much Mana has been spent by the enemy hero, which faceless doesn't really need to spend. If you want to go late game, and you can outfarm an AM, Faceless is a good option.

AM will absolutely rip through your intelligence heroes. He struggles against beefy tanks with high armor, and can be rendered useless with a good silence. Remember that his name is "Anti-Mage" and not "Fucking bulldozer." The reason he was a strong pick in the competitive scene for a long time was because the strongest competitive heroes were those who relied on mana and spells. He has become less of a rock avalanche now that there are heroes than can push down his lane and stand up to him in team fights.

Any invis hero < Slardar
His ultimate which has a very short cooldown provides a damage amplification as well as true sight. Slardar is an excellent way to counter any invis hero.

Tidehunter < Brewmaster
Tidehunter is basically a walking ultimate. He is much less useful if his ultimate is rendered useless. Using Brewmaster to counter Ravage isn't easy, but its possible. While Brewmaster is casting his ultimate and shifting into three brewlings he can't be damaged by ravage. You can counter Tidehunter if you can initiate on him and get him to use his ultimate on you instead of the whole team, and you might have a shot at negating the damage entirely. If you do get hit, just shift into brewlings, which spawn with full health. Once you are back to brewmaster form, you'll have whatever health you had upon shifting, and since you have lots of HP, ravage shouldn't have taken more than 1/3-1/2 so you can still escape. This does take positioning and clever thinking.

Heavy burst-damage teams (Lina, Queen of Pain, etc) < Skeleton King
Skeleton king is a tank with Resurrection. If you can bait the team into hitting you with their burst damage spells instead of the rest of your teammates, you will have essentially negated all of those spells in a teamfight, because if you die, you'll be back shortly, and all of their spells will still be on cooldown, and they will have used a lot of their Mana. Skeleton king is weak when he is focused down on, but when he is a distraction, he's very powerful.

Ursa < ShadowShaman / BeastMaster / Lion

These are all heroes with long stuns (with relatively short cooldowns) and it is really the key to shutting Ursa down: You want to reset his Fury Swipes, and interrupt his Overpower the moment it is activated. The idea is that the moment the CC is activated from one of these heroes - the rest of the team focuses this bear down.

Additionally, BeastMaster's stun goes through BKB. He can use his summoned hawk to scout Rosh (which should be done every time the spell is up) which can be critical to shutting down Ursa: Preventing him from getting Aegis.

Tidehunter, and Engima are also worthwhile picks - however their CC options are on such long cooldowns that they should not be considered good counter-picks.

Shadow Shaman is a great counter to a lot of carries as well. His hex and shackle coupled with a Eul's (core for me) makes him the ultimate chain stunner. Hexing, then shackling, then eul's, rinse, repeat until someone has killed the target. It just ward trap him and do it yourself.

Edit: I also think a BKB is core if you're using him to target a specific carry.

Also - Bounty Hunter makes for a good pick due to his tracking for the exact same reasons.

I generally prefer BH as a counter-pick to Rikki, but Slardar as a counter-pick to other invis heroes like Brood Mother. The armor debuff on Brood specifically helps him tank her when she ults.