Installing Puppet can be a nightmare at times especially if you are doing it for the first time. Error messages are not always obvious and would require some experience to understand. So this is my attempt to explain the errors and suggest the solutions.
Needless to say that step one would always be to ensure that the names are resolving and the puppet client and master can communicate. Also make sure that port 8140 is white listed.
Error 1: err: Could not request certificate: getaddrinfo: Name or service not known
Probable Solution: Puppet client is not able to reach the puppet master. This usually happens when you are setting up a new environment and puppet master's name is not resolvable. If you can, put a relevant entry in your DNS and add a server variable in [agent] section in puppet.conf. Alternatively you can use /etc/hosts to point the client to the master but you'll have to add appropriate entries on the /etc/hosts of both the puppet master and client.
Error 2: Starting puppetmaster: Could not prepare for execution: Could not find a default provider for user
Probable Solution: This happens because of SELinux restrictions. You can fix this by running a "setenforce 0" which will turn off the SELinux. This is required for CA creation only. So you can turn on SELinux after the puppet master creates CA successfully.
Error 3: err: Could not request certificate: Retrieved certificate does not match private key; please remove certificate from server and regenerate it with the current key
Probable Solution: Looks like your certificates have gone bad. You should remove /var/lib/puppet/ssl directory and request for new certs signed by puppet master.
Error 4: err: Could not retrieve catalog from remote server: hostname was not match with the server certificate
Probable Solution: This may happen if you are referring to the puppet master by a wrong name. In other words, the CA is not built to use this name. You can check out the correct CA name in the file /var/lib/puppet/ssl/ca/inventory.txt. You should put this name in the [agent] section assigned to server variable.
Error 5: err: Could not retrieve catalog from remote server: Connection refused - connect(2)
Probable Solution: This is happening because your puppet client is not able to connect to puppet master. One reason might be firewall which is rejecting the packets and the other reason might be that puppet master has died. So you either need to relax your firewall or make sure that your puppet master is always up and running. You may want to use daemontools or god or a similar application.
Error 6: Exiting; no certificate found and waitforcert is disabled
Probable Solution: This usually happens when a new node is introduced in the infrastructure. Issue is that this node do not have the certificate yet and since "--waitforcert" flag was not enabled, it exited immediately. If your puppet master has autosign enabled that just add the flag "--waitforcert X" with X replaced with time in seconds like 60. If autosign is not enabled then you have to sign the cert for the client manually at your puppet master.
I'll add more as I encounter them. Please let me know in comments if I am wrong anywhere. Have fun with Puppet :)