Offshore hosting is nothing but hosting your website , data or application to a far off Datacenter location from your place of origin. There are many reasons for taking your hosting offshore, reasons like hosting copyrighted content, DMCA content , Hacking websites, adult content and warez hosting are few amongst them. There are servers that run/host a website except its located in another country. Many people use this because it is much harder to shut down legally and harder to contact.
Some of the best are:-
Flaunt7
Kamatera
ProHoster
Hostpoco
Hope this information is helpful to you