For a solo lane, you want a hero that's somewhat level dependent, as they'll be getting a lot of experience. This could be someone like an Invoker, Leshrac, or Night Stalker. If this lane is solo mid, ideally they'll also be able to utilize runes well to gank, because they'll have the easiest access to them.
As yx. sort of pointed out, it's much harder to gank through the opponent's jungle than through your own, making the "safe" or "short" lane the one alongside your own jungle. This is typically where a hard carry hero who needs a LOT of items like Anti-Mage or Faceless Void will lane. However, melee heroes in lane are subject to a lot of harass so they'll almost definitely be paired with one or two support heroes like a Dazzle or Crystal Maiden, who will earn a small bit of gold for themselves and maintain lane equilibrium by pulling neutral creeps.
Conversely, the "offlane" or "long" lane is often against multiple opponents and is easy to gank, so you'll want someone there who has a natural escape mechanism ability, like a Windrunner, Puck, Dark Seer, or Mirana, all of whom can be productive just with levels and not much gold, which is exactly what they're likely to be getting in a "suicide" solo lane.
A hard carry is a hero who in lategame with a decent amount of items will be able to deal large amounts of auto-attack damage quickly. They usually have abilities that will make an item on them stronger than the same item on another hero. For example: Gyrocoptor's Flak Cannon, Juggernaut's Critical Strike, and Weaver's Geminate Attack all grant extra damage. These heroes usually have large AGI growth per level or are AGI heroes, which allows them to gain high attack damage, attack speed, and armor from the same attribute.
Some hard carries are also increasingly powerful into the game because they are able to farm at far faster speeds than other carries can match. They have good base stats and/or good mobility and thus can hit a "lategame damage" stage faster than typical carries. Some examples include Anti-Mage, Shadowfiend, and Naga Siren. In addition, there are some unusual hard carry heroes like Spectre, Morphling, and Obsidian Destroyer are able to do so through unorthodox abiltiies. I wouldn't worry too much about that.
The terms aren't very uniform, but "soft carry" or "semi carry" usually refers to heroes that do better with items than the majority of heroes, but not as good as a hard carry. Examples of these might be Windrunner, Mirana, and Stormspirit. While they can potentially carry if they have a good enough early and midgame, with an equal amount of gold in items they are expected to be weaker than a hard carry. They make up for this by having a stronger laning phase/earlygame and having stronger abilities.
Finally, a term that's gaining more traction recently is "midgame carry" for heroes like Razor, Necrolyte, and Krobelus. These heroes deal a majority of damage from their abilities, not from their items, and so they peak in terms of effectiveness when these abilities are maxed, usually somewhere between levels 11 and 16. This is much earlier than a typical carry who just gets better and better as he gains levels. However, they are still fully capable of ending a game in this window of time if left unchecked.
Hope that helps. -.-