This response assumes a perfect translator defined as follows:
The perfect translator is a machine doing it's best work possible. It is not limited by technical errors or construction mistakes and it can be assumed to have had opportunity to study all languages in question thoroughly. Further I assume that both ends of the conversation either uses their own perfect translator or both interface appropriate with it - using what sensors are necessary (i.e, a microphone and camera for humans since they communicate with voices and body language/visual signs).
The answer below thus attempts to answer what languages are not translatable even though there is no lack of proper equipment. (i.e creatures communicating through unusual modalities (chemicals, radio-waves etc.) are not a problem as sensors to detect the modality can be assumed to be present).
Languages untranslatable into certain other languages
A singular real language is actually translated at least twice as part of its normal use.
Speaker A's internal concepts > Language symbols > Speaker B's internal concepts.
As such it requires that both speakers have the same concepts and can express and understand them from the language. Some languages are more apt to expressing certain concepts than other languages because it was developed by speakers wanting to communicate certain concepts.
When looking at a translation between two languages the concepts encoded by the origin language must find symbols or sets of symbols in the target language which gets translated to the same symbols in the minds of listeners.
Even normal human languages fail at this, the meaning of some phrases are very hard to translate and more subtle details are often lost or changed even in very good (human made) translations, thus the phrase "lost in translation".
Considering more alien languages where the original speakers are not even the same species, the underlying concepts of the speakers will differ greatly and the language constructs to express said concepts will be tuned to expressing these special concepts.
To translate between such languages might be very hard to do well - with heavy information loss. If the speakers are too different, communication would be impossible.
Given a perfect translator (a machine doing the best possible) it seems reasonable that languages could be grouped into families where translation would be easy within the family and hard between them. One would expect that families would be grouped depending on the nature of the speakers and their natural environment - more similar creatures would communicate better.
In a society where such translators were commonplace - language families would drift closer to each other, adopt loan words and such - reinforcing the familiarity.
When speaking to an out-of-language-family creature one would expect very general notions to translate better, while emotional and sensory vocabulary would be more lacking. Probably they could both do common tasks like trade using simplistic language constructs that translates good enough only for that activity.
To learn a out-of-family-language might be impossible, or might require large changes in mindset or even sensory augmentations.
Languages intentionally untranslatable
If the availability of artificial translators were common, it is perfectly reasonable for two speakers to converse using a spoken language none of them understands. The translator translates language A into X (understood by none) and the receiver translates it back to A (or to some other language the receiver understands).
Since this is possible, it would be a simple matter to speak in an encrypted language. Each translator installs the same cryptographic-key and produces an encrypted language which can only be decrypted by translators with the same key.
Certain organisations or even cultures could thus have languages impossible to understand by anyone lacking the correct keys.