Snowballing is an unofficial term, describing a concept where once a hero gets enough momentum, they become much stronger (relative to other heroes at that stage) due to that momentum. The general definition of snowballing can be found here.
In Dota 2, snowballing can be used in many different contexts. When a Hero is said to be good at snowballing, this means that when they gain enough of an advantage (usually in early or mid game) they will completely dominate the enemy team, given the enemy does not have a composition to deal with the threat. When a team is said to snowball, this implies that due to a team's certain composition, when they acquire an advantage they are able to keep (and increase) said advantage.
Note that Heroes that snowball are not the same thing as carries. Carries are Heroes that scale very well with items (hence they become quite strong late game when they acquire these items). On the other hand heroes that snowball are ones that are able to maintain and increase their momentum when they obtain it.
A very good distinction to make is that a carry hero is able to turn the tides of a losing team if they obtain enough items and/or drag the game on long enough, where as a snowballing hero is generically unable to turn the tides for their own team if they are doing badly, mainly because a snowballing hero needs an initial advantage (i.e. momentum). For similar reasons, a snowballing hero can fall behind the rest of the team in terms of overall usefulness if they do not gain that crucial advantage. Some heroes can be both a carry and a snowballer (Luna is a good example)
Heroes that are known as Snowballers
Nature of Snowballing heroes
Snowballing Heroes are often niche Heroes, that means they have a few superb strengths but also a few severe weaknesses. These Heroes snowball when they are able to cover their weaknesses to an extent that their strengths shine, and this only happens when they get a strong advantage at the start. These severe weaknesses are what also make snowballers fall behind the rest of their team should they not get their initial advantage. Such Heroes are very reliant on situational abilities and/or items, or the need to verse unconventional lineups. They can also be reliant on either jungling or going into a certain lane (usually mid) which creates obvious counters.
Pudge is widely regarded as the best example of a snowballer. His very high tankiness, combined with his insane burst damage and ability to gank, assassinates enemy Heroes as well as singling them out to allow him to completely dominate over the enemy team. This is further amplified by his Flesh Heap ability which grants him bonus Strength whenever a Hero is killed in his vicinity. The sheer tankiness of Pudge (high starting Strength/Strength gained combined with Flesh Heap) makes it very difficult for the opposing team to bounce back and deal with Pudge when he gains an upper hand.
On the other hand Pudge has glaring weaknesses. He is almost completely reliant on landing his Meat Hook for Pudge to do his job as a nuker since he doesn't have any abilities to attract focus fire to make use of his tankiness. This is due to the fact that his other nukes (Rot and Dismember) require him to be in melee range (that and Dismember is channeling) and Pudge has no mobility combined with a below average movement speed of 285. In other words, Pudge needs to land his Meat Hook very often to bring an opponent into range in order to use his other abilities effectively. Furthermore Rot also deals self damage which makes it even harder to use effectively if you miss your hook and can be taken advantage of by other heroes such as Bloodseeker, Necrophos and Doom. Rot also makes Pudge reliant on a somewhat early Hood of Defiance as it greatly reduces the amount of self damage that Pudge takes.
Pudge's other weakness is that even though he has somewhat expensive mana costs on his abilities, he has very low starting Intelligence (15) and Intelligence gain (1.5) which results into a very low mana pool. This means that Pudge is reliant on Bottle/Arcane Boots just so he has enough mana to use his abilities. This reliance on certain key items coupled with the fact that Pudge isn't a fantastic farmer (he only has last hitting to help him farm and his playstyle means he spends more time roaming and ganking rather than actually farming in a lane) means that kills are his main form of income. If he doesn't get his initial kills in early game, he really falls behind as he completely blows his mana pool after doing a single combo. He is also countered heavily by Observer Ward's as they greatly aid in the opposing team's ability to dodge incoming hooks and also help in preventing Pudge getting his much needed runes.
Pudge is also reliant to solo (almost always in mid) which makes it both very obvious and hence easy to counter. Heroes such as Juggernaut, Omniknight and Bloodseeker can be placed mid to counter a Pudge. Alternately a melee carry such as Phantom Assassin can be placed mid to take advantage of the fact that Pudge doesn't have fantastic lane control and that the Pudge after he obtains his key items will most likely spend more time ganking than harassing the carry in mid.