Private NameServers are DNS NameServers that are associated with your own domain name rather than those used by your hosting provider.
For example, the default NameServers for all accounts hosted by Pay Monthly Hosting are ns1.paymonthlyhosting.co.uk and ns2.paymonthlyhosting.co.uk (along with ns3 and ns4).
If you were to set Private NameServers for "yourdomain.com" you could use ns1.yourdomain.com and ns2.yourdomain.com.
However, in order to use these you would have to make the appropriate changes to your DNS records for these to become effective.
There is no real benefit in doing this other than "branding" reasons. i.e. if someone does a NameServer lookup on your domain it will show ns1.yourdomain.com and ns2.yourdomain.com which will mask the fact that you are using our services.