I completely agree with Jorg & Shayam , its always better & preferable to use shell or winscp to download log files from server.
But still if you want to download log files using curl then below are the curl commands.
Curl command to copy file from one location to another locally
curl file: ///path to source file -o /path to destination
Curl command to download file from remote server
curl -u user:pass -O ftp://remoteurl/download.zip
As mentioned by shayam you can view error log through web console , If you want to download it using curl then below is the command, but its not recommended.
curl -u admin:admin http://host:port/bin/crxde/logs?tail=400 > error.html
View more AEM curl commands
regards,
Ankur