I ain't no Sendmail expert, but I have spent hours configuring it in the past... :-{ Maybe I can add a bit more (approximate) detail to the previous good advice.
First, the easiest thing is not to send mail directly but use your ISP as a mail relay. I presume that you have rejected that and have decided to send out mail directly to the destination using Sendmail or Qmail or something.
You have tried to send mail directly to a mail host which requires the "sender" to have a valid Internic name which is resolvable to something. i.e. they get the senders domain, use DNS to look up the MX records for it and find out if it exists.
Some might even go to the extent of doing a double-reverse lookup i.e. using the IP address returned in the "mail exchange" record to do a reverse look-up to a domain, then using that domain, re-looking at the MX record for that domain (if different). This to make sure that mail CAN get back to the sender and that it looks sort-of legitimate.
So simply having a domain registered may be not enough. You need to
a) get a domain registered (or use a sub-domain of your ISP's domain with their permission and help)
b) get an organisation to be your authority for that domain i.e. manage the records such as your mail exchange records pointing to your mail server either you have to run the DNS server or find someone (such as your ISP) who runs DNS and is prepared to host your personal domain (which is most likely the case if they permit you a sub-domain of their domain e.g. feetyouwell.your-isp.com)
c) ensure that your sendmail (or other SMTP receiver) can receive mail for that domain
If your ISP does not guarantee you a fixed IP address (i.e. if it changes from time to time unpredictably) then the problem is a bit harder. You will almost certainly need to use dynamic dns from a service such as
http://www.dyndns.org/
It is of course easy to check to see if a domain is resolvable, just use nslookup to get all the records for your domain.
I hope that this approximate summary can help you a bit.
All this because of spammers and spoofers - and it STILL does not stop them, just makes your life a bit harder. ;-)