See lines 32, 36, and 54. You have errors in those lines, which should be obvious once you see them. You also forgot to #include <cstring> to use strcmp() but some compilers auto-include it for you.
Additionally, check lines 43 and 52. Are you sure you want those ; there?
Note that line 72, although the intuitive solution, doesn't do what you think it does. :(
EDIT: Check line 26. You're not using sizeof correctly there. Firstly, you're missing ( ), and it returns a size in bytes. In place of the whole hoopla with sizeof, you could've just put 10. The -1 hinders you, as the < is a less than, not less than or equal to operator.
Last, I'd like to clear a few things up about your programming etiquette, if you don't mind. :/
Firstly, never use system(), as it's not only slow and platform dependent, but a massive security hole.
Second, why are you mixing C and C++ I/O, by using printf() and std::cout in the same program?
Third, why are you mixing C and C++ strings?
Foruth, conio.h has been deprecated. Avoid it like the plague. ;)
Finally, you have a ton of headers you don't need. <vector>, <algorithm>, and <cmath> are unneeded here.
I mixed strings because I had no other way of displaying astrixs for the password, which was the only reason I made this to begin with. Is line 36 suppose to be if (strcmp(password, root_password) == 0) ? Im still getting a missing function error on that line.
There's no standard way of creating a backspace, although I'd suggest that if you want to do more fancy console work like this, that you use the PDCurses library. It's portable and far better than using conio.h, as the functionality doesn't change between compilers (!). Just note that it doesn't mix too well with the standard library functions and objects.
Oh... and about the missing function... look carefully at the arguments. I just realized something that I knew I was forgetting about. strcmp() doesn't take strings as arguments; you need to use std::string::c_str() to get a C-string from a C++-string.