The Irken race has their own written and spoken language. The latter seems to be perceived by the listener as their own native language, and therefore has never been documented in its original Irken sounds. It's possible that the common language of the Invader Zim universe is English, since Dib had heard the transmission of Operation Impending Doom II from planet Conventia and was able to perfectly understand it, and in turn he was understood by the Tallest when he made a transmission to them from Zim's own base in "The Frycook What Came from All That Space".
The written form of Irken consists of a series of indistinguishable characters in the show. Jhonen Vasquez had no Irken character chart in mind when he had Irken letters appear on screens and the opening sequence of the show. The Irken subtitles on the DVD and the show itself had totally different characters. The ones on the subtitles consist of the first 20 letters of the alphabet (A-T). The rest of the letters have apparently not been developed, leading to controversy as to whether it's the true language at all. However, it's possible that the remaining sounds (U, V, Y, and Z) either do not exist, or don't need separate letters in their alphabet, as most of their sounds can be achieved with other letters. Because there is no letter for Z, the main character's name is spelled IM on the DVD subtitles. One possible explanation is that a "Z" sound is automatically added before any word that begins with a vowel, but this is mere conjecture, as there is presently not enough evidence to confirm or deny this. Another possibility might be that the correct spelling of 'ZIM' in the Irken language is actually 'SIM'.
The only evidence for the actual sounds of the Irken language are the names of the Irken characters, and what was shown in the episode "Mysterious Mysteries", where Dib recorded a video of Zim and GIR talking in some alien dialect while out of their disguises, possibly Irken. However, this has not been confirmed, and it's equally likely they were just mumbling or talking in a matter humans hear as mumbling.
It's notable that Zim, when pronouncing his own name, seems to frequently pronounce it zîm, with a somewhat rounded I, and not the more common Zĭm, especially when he is alone or talking to himself. If this is done intentionally by the voice actor, it may be an indication of native Irken pronunciation of that vowel - the pronunciation of the I's in Miyuki and Dookie are likewise similar, as is the sound of the double-E in Spleen. Double-O would appear to represent an ōō in sound, as in Dookie and Skoodge, thus perhaps the lack of need for a letter U in their alphabet. The Irken language does seem to accept multiple consonants side by side.
The following is a list, by no means complete, of various Irken names: Invader Alexovich; Invader Chin; Invader Dooky; Invader El; Invader Flobee; Invader Gooch; Invader Grapa; Invader Groot; Hok; Invader Kim; Invader Koot; Invader Krunk; Invader Larb; Invader Lardnar; Tallest Miyuki; Invader Nen; Invader Pesto; Tallest Purple; Invader Poot; Commander Poki; Tallest Red; Rarl Kove; Invader Sklud; Invader Skoo; Invader Skoodge; Skrang; Invader Skutch; Invader Slacks; Invader Slant; Sizz-Lorr; Invader Sneakyonfoota; Invader Spleen; Invader Sploodge; Tallest Spork; Invader Sqood; Invader Stink; Tak; Invader Tenn; Invader Tim; Invader Yoogli; Invader Zee; Zim.
Jhonen Vasquez states that Irkens use all caps "because they're annoying."  This explains why Zim's name is always written as "ZIM" on the official voice transcripts. It can be assumed that all other Irkens write their names in the same fashion.
The names Miyuki, Skutch, and Zim all contain letters that appear to not exist in Irken. Miyuki could theoretically be spelled Miooki to achieve the same pronunciation, and a theory on Zim's name can be found above. Skutch might be perhaps Skatch or Sktch; it also could possibly be a variant and/or related name to Skoodge, just as English Mary and Marie are variants on the same name. It's also possible that these names are not native to the Irken language. Since very little is known about Irken naming conventions, nothing can be said with any certainty. It's notable that Irkens do have a diminutive attachment for names in their language, as evidenced in "Hobo 13" when Tallest Purple addresses Table-Headed Service Drone Bob as "Bobby."
One theory that might explain that Irken has no specific alphabet is that Irken is based on pictograms, albeit ones that have evolved over millennia. Written Irken might be formed of thousands of symbols, much like Chinese writing. However, there is no evidence to really support this argument. More than likely, Irken has no specific alphabet because Jhonen Vasquez had not developed an official Irken alphabet.
Other evidence for the vocabulary and sentence structure of Irken might perhaps be found in Zim's sometimes odd turns of phrase. (i.e. "arm-control nerve", "noise-tube", "cleansing-chalk" et al.) It's possible that he is using what might be correct terms in Irken, but which translate badly into English, or it could be due to his questionable sanity.
Since Irkens appear to be the dominant species in the known universe, its possible that other alien races speak English as a result to be able to communicate with them, leaving to the possibility that Irken IS English, but with a different written alphabet. This would explain why Dib did not have to translate the Irken transmissions, and why Zim had no need to learn English.
More evidence supporting this theory is found in the episode "Abducted", when a transmission between Zim and the Almighty Tallest was interrupted by the doorbell ringing; on a nearby display screen, an image of the doorbell is shown along with a notable amount of Irken. However, there is one English word on there: "Doorbell". However, it should be noted that Zim was trying to read the word with some struggle, implying that he was attempting to learn human writing little by little which he apparently achieved at some point before the episode "Zim Eats Waffles" as he was seen reading the newspaper.