Summary: In your case I would recommend to use execvp.
To find out the differences between the exec* functions you should read the documentation:
https://linux.die.net/man/3/exec
https://linux.die.net/man/2/execve
The difference between execl* and execv* is the argument passing. execl* require a list of arguments while execv* require a vector of arguments.
A list of arguments is useful if you know all the arguments at compile time. In your case the arguments will be entered by the user and you have to construct a vector of arguments at run time, so you should use one of the execv* functions.
The functions with suffix p use the PATH environment variable to find the program (e.g. "ls"), without this you have to specify the full path (either absolute or relative to the current directory, e.g. "/bin/ls"). Using PATH is what shells normally do, so this seems to be the correct choice for you.
The functions with suffix e allow to specify the environment for the process. For simplicity I would not use this in your case.
This leads to the conclusion: execvp
Of course you could also use system (instead of fork/exec*/wait*) as mentioned in vladxjohn’s answer, but in this case you would merely use a shell to interpret your command instead of implementing a basic shell.