I am going to guess that you can receive emails with no problem, but you are unable to send any outgoing emails. I believe your problem is related to the Internet Service Provider for each computer. Your computer users Port 25 to send emails through your Outgoing Mail Server, or SMTP (i.e. mail.yourdomain.com), from your computer.

Many ISPs today are blocking your ability to send outgoing email messages through Port 25 because of the increase amount of spam. I believe Port 25 is not blocked on your friend's computer, which is why you are able to send your outgoing emails with no problems at all from that computer, but Port 25 is blocked on the computer you are not able to send your emails with. Remember, it is your Internet Service Provider (or ISP) that does this.

Many ISPs will block Port 25 and say that you can't send any outgoing emails, but some will restrict Port 25 so that you can't use your own personal Outgoing Mail Server (i.e. mail.yourdomain.com), but instead you will be provided with the ISP's Outgoing Mail Server because it gives them more control over spam levels. For example, for months I was able to use my own personal Outgoing Mail Server (i.e. mail.mydomain.com) until Charter Communications, which is my Internet Service Provider, restricted Port 25 and I was forced to use Charter's Outgoing Mail Server, which is smtp.charter.net.

There are some tests you can run on each computer through command line that will tell you if the computer's Port 25 is blocked, but in most cases, if you see that you can receive incoming emails but not send any out (assuming you input your email account's login info correctly in your email client), it can be an issue related to the restriction or blocking of Port 25 by your ISP. If you tell us the names of Internet Service Providers for each of the computers you are trying to send outgoing emails with, I'm sure we can tell you which ones are known for restricting or blocking Port 25 and which may require further research.

Again, based on the information you have given me this is my best guess. I may be wrong, but since many of my web hosting clients experience the same problems with their Internet Service Providers, I have become accustomed to hearing about the restricting and blocking of Port 25 by ISPs.