Prologue
The power to understand any code... At the power to understand any code that has a meaning.
We can start by defining "symbol". Symbol would be any series of marks or sounds that represents something. Now a "language" would be an encoding system that uses those symbols to convey meaning. We would call each set of symbols that has complete meaning according to a particular language a "message". Since we are counting both visual and auditory symbols... a message can be spoken, singed, written, drawn, etc..
These are rather general definitions, so if you want to add constraints please edit your question, but for what you have said…
even the "dit-dah" of telegraph
It seems that any artificial language is fair play.
Another thing I would note is that a message may have multiple meanings, and the character would understand them all at once. I do not interpret this power as being able to read the intention of the author of a message, but to be able to understand any information conveyed by the message itself and its context (for example the for of the letters on a manuscript may give away information about the author).
Note: I will be refering to the character as "the character", not taking any gender assumtions here.
On the limitations of the power
It is interesting that the character just understands. So we are not talking about a universal translator per-se, but with some training/focus this character could translate anything to a language he knows.
That's a notable distinction, the character may know a few languages (as in: he can use them to communicate) but will also understand messages in languages that are unknown.
I don't see the influx of information as annoying, but the information that the character picks up may have emotional effects and may challenge sanity. Stuff like reading micro-expressions, being able to tell at a glance - or by the pitch of the voice - what is a lie.
I want to note that I don't equate the characeter to a supercomputer. I mention below that the character would be able to understand any programing language, but the power is not to make computations in his mind, nor to simulate the behavior of software.
I want to make emphasis in that the power is understanding. The same way that understanding how a hashing algorithm works doesn't grant a person the skill to automatically come up with a password that produces a particular digest made with the hashing algorithm. Also, the same way that understanding how integration works doesn't grant a person the skill to answer a defined integral in an instant.
Cryptography
So, the character understands a message even if it is in an unknown language and even if that language is artificial...
The character can read all cryptography
Any message encoded in whatever convoluted code - as long as no information is lost in the encoding (as happens in hashing) - can be understood by this power.
Note: No, reading the **** as the password won't work. The password is not there.
Wireless communication
Thankfully the power is limited to vision and audition; otherwise the amount of info coming in would be too much. Yet with some ingenuity a whole lot of stuff is possible, all that is needed is to convert a signal to a visual or auditory representation...
For instance, tune a radio receiver to some seemingly uninteresting frequency and you would pick no strong signal, just noise. But there is information there.
Do it with wifi frequencies, cellphone frequencies, etc... it could spy any wireless communication, even under encryption.
Math
A mathematical expression is a series of symbols; in fact a large portion of math can be understood as symbol manipulation. In particular, any expression that can be simplified or any equation that can be solved without breaking the equivalence would reveal its meaning.
For instance when presented with “x^2+y^2=1, (x-2)^2+(y-1)^2=4” the character would know that those are two circles, one with center in (2, 1) and 2 units of radius, the other with center in the origin and radius 1. That’s because that’s what it means.
We say that the character understand any language, so the character understands math too, and so when presented with an equation that correlates two variable, the character understand that relation.
If the character understands the relationship of the variable, the character understands how they behave relative to each other. The character could – for example – know just at glance if a function is strictly ascending or if the relationship is exponential, or whatever the variables is independent of each other.
The character would understand the math expression it simplest form. So x = 4/8 would be evident that it means that x = 0.5 because that what it means. So the character did actually gain some processing power. To make it clearer, you may not be inmediately aware that x = 321321351436584/13388389643191 means that x = 24, but well, that's what it means.
So when presented with x = integrate from 0 to 10 {35 + sin(7)^0.25 dx} the character could know that x = 350.591 because that what it means (what is at the right hand side is a constant expression after all).
When presented with f(x) = (x^2+3)/(x+1) the character would know that the function has an asymptote at x = -1, that the intervals (-∞, 3), (-1, 1) it is an increasing function, and in the intervals (-3, -1) and (1, ∞) it is a decreasing function… and so on…
Puzzles
Many puzzles have the characteristic that they present all the information needed to solve it (ie. that they don't require investigation or prior knowledge). For example a Sudoku puzzle presents all the information required to complete it.
We could understand this form of compression, and thus a kind of encoding. That implies that the character would be able to use the power to know the solution of the puzzle.
Paradoxes
The character is presented with the following:
A: B is True
B: A is False
This is an example of the liar paradox. As any good paradox it is problematic...
Whatever or not this paradox is ultimately meaningless is debated. See: Logic nonsense/paradox.
What the paradox says is A means that B is true, and B means that A is false. So if A is false, it must be false the B is true... ergo, B must be false, which means that it must be false that A is false... ergo, A must be true, meaning that B must be true... ergo, it must be true that A is false... if we continue the deduction in this fashion it will never end.
One alternative is to reduce the expression to A: A is false. It is the simplest form. If A is true, A must be false, if A is false it must be false that A is false, ergo... A must be true. It just made the loop tighter; it didn't lead to any ultimate meaning.
Another option is to use Kleene's logic, under which the solution is that A and B are unknown. Yet, that's not satisfactory either.
My favorit alternative boils down to A = false and B = undeterminate... but what does that mean?
Whatever or not this will put the character in vegetative state depends on how the power works. Maybe the character can compheend the paradox meaning, as granted by God. On the other hand it could be better to keep the power off unless the character is decided on using it... just in case it can't survive a paradox.
Programing languages
The character can read all programing languages from A+ to ZZT, including Brainfuck and LOLCODE. Just give this person the code base for any software and he will tell you any detail of how it works... give the character a memory dump and will tell you what is going on.
I would suspect that this person doesn't have the training to simulate software in his mind, as that is not part of the power... yet, the character could be able to pick up defects and vulnerabilities of the software.
This power useful to answer questions of the form "what happens if I do XYZ with the software?" once the character understand the code of such software.
Now you have a great spy/hacker.
Forensics
Yet information is not only encoded deliberately. For example people may use body language unconsciously, and the character understands it all, even from aliens or animals.
And even more subtle things may encode information... for instance I leave an object in a visible place to remind me to pick it up... the character understand this too! The character can read intention in the placement of objects, as this can be understood as a visual language.
This implies that the character has the uncanny skill to read a person, from his body language to his selection of clothing. Or read a room, in the sense that he could understand the implications behind the furniture and objects... everybody can do this to some extent, what I say that this would be natural and no piece would be left unexplained.
Now you have a great spy/hacker/detective.
Also as mentioned by a4android this power understand dead languages is useful for archeologists.
DNA
Show the character a genome sequence. Similar to how the character could interpret the readouts from SETI... the character could read the genome... "That's an antibody, That's an enzyme, that's a biological weapon, that's a message from the andromeda empire that says 'squids are stupid'..."
Consequences
What are the consequences of this being? Well, depends on what will do with this power and whatever or no other people notice. I think the strongest point would be to use the character for espionage...
If you have the character doing espionage work - something like hooking the character up to ECHELON - he would know of a lot of crazy stuff that goes on around the world. You then ask you could the character about something, and the character would just know... will tell you if the character doesn't think that this kind of espionage is unethical, and you are not torturing the character enough.
Unless the character wants to end up in that situation, I would expect that hiding the power is a better idea. Sure, there are better avenues as diplomat or plain old translator, which would be ok as long as nobody suspects about the power.
Of course, some other employment avenues are in code breaking, genetics, ect...
In particular puzzle games such as FoldIt or Eterna could have the character solving the structure for proteins to tag viruses. Perhaps will not get the cure for cancer, but the cure for AIDS.
EXTRA: The gaining knowledge interpretation
Under this interpretation, when presented with a message the character would not only what the message means, but what it refers to. This means that any knowledge associated with the message is granted.
For example: When the character sees the messages "The andromedan empire", the character would know what it means, but not whatever or not it is real, or how it is. Yet, under the gaining knowledge interpretation, the character would know what is there to know about the andromedan empire.
**The meaning of life? **
We use the word "life" as if we knew what it actually means... another problematic word is "person". This are simple concepts but the boundaries are not clear. Would the character gain ultimate understanding of the meaning of life upon being presented with the word?
hacking the power to gain knowledge
When presented with "Did Napoleon die in 1821?" The character would understand the question, that doesn’t imply that the answer is revealed. But, again, with a little ingenuity, it could work... just write: "The year of the death of Napoleon" and the understanding would be granted.
The implied consequence of this is that the character can understand everything as long as there is a way to refer to it. Write "Exo-planets with life" and now you understand.
The non-implied consequence is the storage capacity. Usually we would say that this goes to Semantic memory, but the OP says "understand" not "know". So a simple solution for preveting the head of the character to explode※ is to say that the meaning of the messages is in short-term memory... distracted and puff! The meaning of life is gone.
As per meaning too big to comprehend all at once, well, they don't have to come all at once. Under the idea that a paradox may leave the character vegetative, an overwhelming concept may leave him in awe for some time.
※: Quite literally. Something like "All the stars in the universe" would be too much for the poor brain.