Ok. First off, being able to PING someone and send them e-mail are not specifically related. As Answer1 states, a lot of places are blocking ICMP packets. Many firewalls now suppress ICMP from passing through it to reach its protected servers, due to DoS attacks. Yet, just because you can't ping a domain doesn't mean you can't e-mail it either.
Now, if the domain was down (their servers were offline, their firewall was down, etc) then yes, you couldn't PING them or e-mail them. But, let's not focus on that. That is an extreme reason for your problems.
Now, if you saw a misconfiguration in the MX records doing an NSLOOKUP, then that could be the problem.
Ok, make sure their MX records are correct. Do another NSLOOKUP against your DNS servers (either your own DNS servers, or the DNS servers your ISP provides), and check if the FQDN and/or IP address supplied for this other domain is correct. If you know what the IP address for this domain is, then is the IP in the MX record right?
Another thing to check is try and traceroute to this IP address also. Again, it might not work, but you can try.
Also, try and go to a web site on that IP address. Just put the IP address you get from the MX record in your web browser (with HTTP:// in front of the IP) and see if it points to the other company.
Also, check your mail server, see if it has the correct DNS settings. Make sure the DNS server can resolve the IP address to the FQDN in the MX record.
Lastly, call the other company and see if they are having trouble.