An int can store values from approximately -2.1 billion to +2.1 billion (exactly -2,147,483,648 to 2,147,483,647). This range comes from using 32 bits with one bit for the sign. You can check the exact limits on your system with #include <climits> and printing INT_MIN and INT_MAX.
If your program needs to count something larger than 2 billion, int will not work. You will get overflow, which I will show you next.