In C the function strcat does not create a new character array containing concatenated strings. It appends characters from the second string to the first string of the first character array provided that it has enough elements to store the new characters. Otherwise the function will try to overwrite the memory beyond the character array that results in undefined behavior.

So a valid use of the function in the first program can look the following way

#include <stdio.h>
#include <string.h>

int main(int argc, const char *argv[]) {
    char s1[11] = "12345";
    char s2[] = "abcde";

    strcat(s1, s2);

    puts(s1);
    puts(s2);
    return 0;
} 

In this program the character array is declared as having 11 elements. Thus it is able to accommodate the appended string "abcde".

In the second program there is an attempt to modify the string literal pointed to by the pointer s1. String literals in C and C++ are immutable. Any attempt to change a string literal results in undefined behavior even though in C opposite to C++ string literals have types of non-constant character arrays.

From the C Standard (6.4.5 String literals)

7 It is unspecified whether these arrays are distinct provided their elements have the appropriate values. If the program attempts to modify such an array, the behavior is undefined.

So in the second program you again need to use a character array with enough elements. For example

#include <stdio.h>
#include <string.h>

int main(int argc, const char *argv[]) {
    char s1[11] = "12345";
    char* s2 = "abcde";

    strcat(s1, s2);

    puts(s1);
    puts(s2);
    return 0;
}

Or you could use either a Variable Length Array (VLA) if the compiler supports them or dynamically allocate an array. For example

#include <stdio.h>
#include <string.h>

int main(int argc, const char *argv[]) {
    char *s1 = "12345";
    char* s2 = "abcde";
    char s3[strlen( s1 ) + strlen( s2 ) + 1];    

    strcpy( s3, s1 );
    strcat( s3, s2 );

    puts(s1);
    puts(s2);
    puts(s3);

    return 0;
}

Or

#include <stdio.h>
#include <string.h>
#include <stdlib.h>

int main(int argc, const char *argv[]) {
    char *s1 = "12345";
    char* s2 = "abcde";
    char *s3 = malloc( strlen( s1 ) + strlen( s2 ) + 1 );    

    if ( s3 != NULL )
    {
        strcpy( s3, s1 );
        strcat( s3, s2 );

        puts(s1);
        puts(s2);
        puts(s3);
    }

    free( s3 );

    return 0;
}

I wonder why no one mentioned snprintf() from stdio.h yet. That's the C way to output multiple values and you won't even have to convert your primitives to strings beforehand.

The following example uses a stack allocated fixed-sized buffer. Otherwise, you have to malloc() the buffer (and store its size), which would make it possible to realloc() on overflow...

char buffer[1024];
int len = snprintf(buffer, sizeof(buffer), "%s %i", "a string", 5);
if(len < 0 || len >= sizeof(buffer))
{
    // buffer too small or error
}

Edit: You might also consider using the asprintf() function. It's a widely available GNU extension and part of TR 24731-2 (which means it might make it into the next C standard). The example from above would read

char * buffer;
if(asprintf(&buffer, "%s %i", "a string", 5) < 0)
{
    // (allocation?) error
}

Remember to free() the buffer when done using it!