Since C++11 we can use auto a = 1+2 instead of int a = 1+2 and the compiler deduces the type of a by itself. How does it work? Is it slower during compile time (more operations) than declaring the type myself?
auto is asking the C++11 compiler to make some limited kind of type inference (look into Ocaml if you want some more sexy type inference language). But the overhead is compile-time only.
If you replace auto a=1+2; with int a=1+2; (both have the same meaning, see answer by simplicis) and if you ask your compiler to optimize (and probably even without asking for optimizations) you'll probably get the same machine code. See also this.
If using GCC try to compile a small C++11 foo.cc file with g++ -Wall -fverbose-asm -O -S foo.cc and look (with an editor) into the generated foo.s assembler file. You'll see no difference in the generated code (but the assembler file might perhaps change slightly, e.g. because of metadata like debug information etc.)
If you are concerned about slower compile-time I guess that using auto is not a decisive factor (probably, overloading could be more costly in compilation time). C++11 is nearly designed to practically require a lot of optimizations (in particular sophisticated inlining and constant folding and dead code elimination), and its "parsing" (notably header inclusion and template expansion) is costly.
Precompiling headers and parallel builds with make -j (and perhaps ccache or distcc) might help in improving the overall compilation time, much more than avoiding auto.
And if you wanted to systematically avoid auto (in particular in range-for loops like std::map<std::string,int> dict; for (auto it: dict) {...}) you'll end up typing much more source code (whose parsing and checking takes significant time) with more risks of error. As explained here, you might guess slightly wrongly the type, and expliciting it (slightly wrongly) might slow down the execution of your code because of additional conversions.
If using GCC you might pass the -ftime-report to g++ and get time measurements about various GCC passes and phases.