Others have mentioned Meowth's ability to speak being something he trained himself to do (with references to the episode which describes it). The other pokemon who have been shown to 'speak' tend to fall into two categories: Legendary pokemon and Psychic pokemon. Some of them, notably Mewtwo, are both.
Initially, Mewtwo was the only non-Meowth Pokemon who could speak, and it was implied that it was possible only because of his incredibly potent psychic abilities and his long exposure to humans. Notably, his 'speech' is also generally shown as telepathy, not verbal speech.
In later movies, notably Kyurem vs. The Sword of Justice, many other Pokemon are able to make themselves understood. That said, most of the time we hear them speak it is to one another. This is not unusual - we see even in the early episodes of the anime that all pokemon understand each other, even if their vocalizations are seemingly simple repetitions of their names. We do, however, see evidence that Kaldeo and Kyurem can make themselves understood by Ash.
It seems that Legendary Pokemon are capable to making themselves understood by humans, but not all Legendaries we have seen use this capability. It's impossible to tell if those we do not see speaking are incapable or simply unwilling.
It seems likely that all Pokemon in the anime could be capable of human speech, but the vast majority are unable or unwilling to put in the time and effort to do so. Legendary pokemon, on the other hand, seem to either be able to communicate with humans innately or at a minimum have a much easier time learning how to do so.
