Will the Mailing List package work on my site?
To run the mailing-list package, your server/website must be able to already send out emails. It's pretty easy to test if emails can be sent from your site. Try adding a "Form" block somewhere on your site, with your own email address as the recipient, and do a test submission. Did you recieve the email? If so, then the mailing list package should work fine. If not, then the first thing you should try is enabling SMTP emails for your site (see below).
Enabling SMTP emails:
The most common reason why emails aren't sent is that your server doesn't support the default php mail() function. In that case you need to enable SMTP emails. The mailing list package will not work on a server that doesn't have either of these available (like when your developing locally on a PC). If you have SMTP available (as most hosted servers do), you can switch your site over use SMTP mail through your /dashboard/settings/mail page. You may need to get SMTP connection information from your hosting companies support staff. Sometimes SMTP will work without any authentication information needed. Try reactivating the send process for your previous mailing to see if SMTP worked. If it still doesn't then you may have just gotten the connection information wrong.
If you're still not recieving emails, then you need to figure out if they are infact being sent.
Does the mailing detail page give a status of "Completed"?
If so, go to your /dashboard/reports/logs/ page, and see if it logged the sent emails. If concrete is working correctly, then you should see a record there for the last emails it sent (by default it only logs the first 5). If it is not there, then there is probably a problem with your core. If the emails are shown there however, then there is a problem occuring outside of the scope of the mailing list package. In this case it's possible that emails from your server are being treated as spam. You should check your inbox's spam folder. If it's there, then there is lot of documentation on the web on why this might be happening (SPF records, IP blacklists, etc), but unfortuantely it's not something directly related to the mailing list package.
Does the mailing detail page give a status of "Interrupted"?
If it does, then it should also give a small gray error message that should tell you what the problem is. If you're still having trouble, then you can request more help in the mailing-list package's support forum: http://www.concrete5.org/marketplace/addons/mailing-list/support-and-feature-requests/
If you do open a new support request, please state whether you've tried using SMTP emails, the last mailing status message you received (with the small gray text if it's shown), and whether the emails are being logged on the reports page.
Does your server seem unresponsive during the send process?
Sending out mailings to thousands of subscribers can consume a lot of your server's system resources. On shared servers this may result in sluggish response times when trying to load your site while the mailing list is running. However, the mailing list package does have some throttling options built in, which you can set to prevent the send process from consuming too much of the system resources at a given time. If you are noticing issues like this, navigate to the settings page under the mailing list section of your dashboard. You'll want to experiment with the two fields on the "Send Rate" line. The number that you send at once largely depends on the processing power of your server, so there's no hard fixed number. For starters try to break the send process up into chunks so that after sending out maybe 10% of the emails at once, and you can make it pause for 30 seconds to a few minutes between each burst of emails. That will give a window for the server to resume processing other non-mailing related requests to the server. If you have it pause for too long then it's possible the mailing will run into some max execution time limit set within the PHP configuration. If this happens though it's not a big deal, since you should be able to just resume the mailing to restart it where it left off.