In reply to:
I Just set up my domain name and bought hosting with Dreamhost. I have been trying, unsuccessfully, to send emails through Entourage X for mac. Somehow, I can receive emails, I just can't send any.
What' I've tried
-Setting up a POP account.
-Setting up an IMAP account.
-Overriding the SMTP default of 25 and setting it to 587.
-Selecting the "SMTP requires Authentication" option.
-Un-Selecting the "SMTP requires Authentication" option.
-Clicking on "SMTP requires secure connection (SSL)"
-Un-Clicking "SMTP requires secure connection (SSL)"
This is getting incredibly frustrating. Any advice would be greatly appreciated.
My first suggestion is that, given all the different things you have tried, it is quite likely that you have gotten your setting pretty significantly "borked". I'm not saying that you have not tried these things, but trying each of them in varying combination can very easily lead to *never* getting the right configuration unless you have very carefully tried *every* combination (and it is very easy to "miss" one configuration or another when going the "reial and error" route). 
In reply to:
...yes and yes. I also followed the instructions at: http://wiki.dreamhost.com/KB_/_Email_/_Client-Specific_Setup_/_Mac_OS_/_Mac_OS_X#Microsoft_Entourage
to the absolute letter with no success. I also tried sending out of Thunderbird with no luck, but surprisingly I have been able to send out of Apple mail quite easily. Too bad Apple mail is so basic, otherwise I'd forget about Entourage and Thunderbird in a heartbeat.
Even following instructions "to the letter" can leave you wanting, depending upon your ISP settings - the instructions cannot cover *every* potential situation.
The fact that you *can* send and receive using Apple mail clearly means one of two things - either the settings that Apple mail is using are *correct*, or Apple mail is actually "sending" using a different server (though I trust you have triple-checked, and eliminated that possibility).
This leads me to believe that the settings in Apple mail are likely correct; you just need to make sure that you duplicate *each and every one* of them in you client of choice.
I can't "walk you through" Entourage, as I don't have it available. However, Thunderbird most definitely *does* work with DreamHost mail (I use it constantly), though I can't give you the precise settings as yours may well vary depending upon your ISP.
For example, for *my* ISP (using POP3, which I prefer), the settings are as follows (using the ThunderBird field descriptions):
Basic Account Server Settings (receives mail)
Server Type: POP Mail Server
Server Name: mail.mydomain.com
Port: 110
User Name: myemailusername
Security Settings:
Use Secure Connection: "never" radio button *is* checked
Use Secure Authentication: checkbox is *not* checked
(Other server settings are preferences - set as you wish)
Under the "Advanced" button in the lower right section of the lower "Server Settings", these setting are for your preference determining how the client handles you inbox folders and whether this server is included when you "get new mail" - set as you desire.
That handles the "receiving" of mail - now onto the "sending" (Outgoing Server).
Look in the left pane of the "Account Settings" box, at the bottom of the Accounts list, and click on "Outgoing Server".
Here it can get a little tricky if you have multiple outgoing servers setup, as I do, so you will have to either "edit" your existing outgoing server if this is the only email account you are using, or "add" the new DH server if it is to be an additional outgoing account.
Either way, my settings end up as follows:
SMTP Server
Description: Whatever you want to use to describe the server
Server Name: mail.mydomain.com
Port: 587 (default is 25)
Security and Authentication:
The checkbox for "Use name and password" *is checked*
UserName: emailusername (same as in the POP setup)
Use Secure Connection: The "No" radio button *is selected*.
This complete the SMTP setup.
Next, *IF YOU HAVE MORE THAN ONE OUTGOING SMTP SERVER SETUP IN THUNDRBIRD*, in the Accounts coloumn (left pane of the "Account Settings window", click the "Description" of the new account you just created/edited (the "Top" level of the settings (which may indicate only "mail.yourdomain.com", depending upon any description you may have used, or not used). When the next screen opens, edit as desired but *make sure* you select the "Outgoing Server" you just defined from the pull-down menu if you have defined more than a single outgoing server.
You can set Manage Identies if you wish, *after* you have this thing working. .