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runtime is the time in which process is executed (have CPU).
If in this time any error happen said runtime error.
and runtime library is library which not linked at compiling time, but, it used when process is in runtime mode.
Talking of "importance" it depends on cotext.
runtime libraries you wouldn't find by disassembing the executable file,
it also means that exe would "weight" less by size of the runlibs - calling holders space. It also whould be important - that in part of running code of runtime libraries the is no garantie that the code exist at all in some paricular machine,
if libs included in your file -you may don't care about it. If it's runtimes libs - you probably should check out if in particular machine there are needed runtimes libs.
And make case-hanldler on the case that there are no needed libs (for example displaing some text of what files (dlls, sys etc.) you need to run your app. and then safely exit)
The same about runtime errors - you should keep in mind that it's kind of errors that your compiler can not catch, yet they can appear while executing the prog.
For example in some case there can be some page falt or div by zero which you can not predict for sure. Yet if there are such a possibility you should case what your app. should do if that's the case.
runtime is the time when your compiled code gets executed . runtime errors - errors which were not detected at compilation time but appeared during run of compiled code.
runtime libraries - your compiled code needs other codes to execute . Let say if you use printf() function in your program , compiler should add the binary of the printf() to your program binary at compile time (static library linking) or put reference to that function within binary (dynamic library linking) - symbol pointing function to be called.
Some runtime errors could arise from improper usage of C such as poor casting from one type to another (mentoined by The Svin ) , buffer overflows when buffer bounds are not checked before access it, improper pointer usage ...
It is recommended to use lint or similar code consistency checker tool o your source code. It will provide much more warning than your favorite compiler . Also there are safer libraries which replace your compiler's to avoid flaws in your program.