Almost all Pokemon in Pokémon, who can say only their names or parts thereof. With a little careful listening to regular cast members (such as Pikachu) in the original Japanese, though, it's possible to make out patterns. Ash (Satoshi) is "Pikapi", while Togepi, during the seasons that Misty (Kasumi) was with them, was "Pipipi." Likewise, Misty herself is "Pikachupi" and Team Rocket was "Pipikachu." Here's a list. The human characters understand them just fine; actions speak louder than words, after all.
Similarly, Mone-chan in Yumeria only says "Mone" or parts thereof. (It is unknown whether or not "Mone" is her real name; it was (unsurprisingly) the only thing she said when asked what her name was.)
Ryo-Ohki, the Weasel Mascot in Tenchi Muyo! can only say "Miyaa," but is clearly an intelligent being capable of communicating complex ideas to the people around her. This is mainly due to the fact she is still very young. In Tenchi Muyo GXP however, she is fully capable of speaking clearly. (Also The Voiceless)
Likewise Mokona from Magic Knight Rayearth can only say "Puu." While he may not seem to act like it and is accorded little respect, Mokona is arguably more intelligent than anyone else in the cast — but the details are a spoiler of monumental proportions. (Also The Voiceless)
Daikichi Komusubi from Eyeshield 21 communicates mainly through grunts and short sentences. It's said he speaks in pawa-go ("Powerspeak"), which only strong men, like his "sensei" Kurita, can understand (though oddly enough, Mamori seems able to understand him as well).
This trope is made fun of in Full Metal Panic? Fumoffu when Sousuke (using a suit of Powered Armor whose voice filter turns every word he says into "fumoffu") and a man dressed in a pony mask (don't ask) have a discussion consisting entirely of "fumoffu fumoffu" and "pony pony." Kaname then yells at them for reaching an understanding just by exchanging unintelligible words.
The series does play it straight however, in the fact that Kaname can actually understand Sosuke in the suit.
Isn't she wearing a comm headset in those scenes?
She seems to be able to translate the One-Liner when he rescues her from some Yakuza. She didn't have the headset then.
The AIR TV series featured a cute dog creature named Potato that "spoke" exclusively using the word "piko" and, occasionally, "pikori." Her owner, Kano, could understand him perfectly, but nobody else could.
Young I-Pin from Katekyo Hitman Reborn!. She's supposed to be speaking Chinese, but no one else among the group can understand her. Aged-up I-Pin is somewhat easier to understand, having "taught herself" to speak the language (still not perfectly, but she's getting there).
The Minicons in Transformers Armada speak in beeps. Towards the end of the series, they start actually talking. It's not clear how that works.
The other characters' ability to understand them also varies. The humans get it most of the time, but Jetfire can't understand them at all and most of the others only vaguely know what they're saying some of the time. It's possible they just have really, really thick accents.
Sweden from Axis Powers Hetalia speaks in a rather thick accent (Tohoku accent in the original Japanese, which translated into English is a sort of grunt speak), so sometimes readers have difficulty following his pattern of speech. Strangely when it shows him thinking, his thoughts are perfectly comprehensible.
.hack//SIGN has Macha, whose lips move and the other characters all react as though she's speaking normally... yet no sound can be heard.
That's supposedly because she's using the text chat, which isn't displayed in the anime, for slightly obvious reasons.
Penelope/Chibimaru from Hamtaro can only say the words ookyoo or ookwee; her "older sister" Pashmina/Mafura is always on hand to explain what the youngest member of the group says.
Lum's Mother from Urusei Yatsura, due to her lack of knowing Japanese, speaks in her native language, which humans can only interpret as gibberish (shown quite clearly in the manga, using various symbols in place of text). Naturally, Lum and other aliens don't have trouble understanding.
President Aria from ARIA is described as having human-level intelligence, but speaks only with the syllables "pui" and "nyu." Similarly, President Maa also only says "maa" (and was named so named for that very reason). Somewhat justified in that they are both cats - they're special breeds that can understand human speech, but they aren't capable of producing it.
Jinpei, the original character, spoke and acted like the smart-ass 10-year-old that he was. Not something the moral guardians wanted to see in an Americanized cartoon. (Joe was the one with the blunt way of speaking.)